ON SOME PROPERTIES OF THE FUNCTIONS OF ONE OR TWO REAL VARIABLES.

Tiberiu Popoviciu,
Former Student of the École Normale Supérieure

Request on March 8, 1933.

Introduction.

In function theory, we seek to delve deeper into the study of very general functions that, in some way, resemble known functions. The simplest functions are polynomials, so it is quite natural to study functions to which certain properties of polynomials apply. It is this kind of problem that we address in the first part of this work.

\left[x_{1},x_{2},\ldots,x_{n+1};f\right]=\frac{\mathrm{U}\left(x_{1},x_{2},\ldots,x_{n+1};f\right)}{\mathrm{V}\left(x_{1},x_{2},\ldots,x_{n+1}\right)}

Or $\mathrm{U}\left(x_{1},x_{2},\ldots,x_{n+1};f\right)$ is the order determinant $n+1$ whose general outline is $1x_{i}x_{i}^{2}\ldots x_{i}^{n-1}f\left(x_{i}\right)$ And $\mathrm{V}\left(x_{1},x_{2},\ldots,x_{n+1}\right)==\mathrm{U}\left(x_{1},x_{2},\ldots,x_{n+1};x^{n}\right)$ .

There $nth$ divided difference of a degié polynomial $n$ is constantly equal to the same number; this number is zero if the polynomial is of degree $n-1$ We examine the functions whose nth divided difference is bounded. We also consider the nth total variation (or the total variation of order $n$ ) of a function, which is by definition equal to the upper limit of the sum

\begin{gathered}\sum_{i=1}^{m-1}\left|\Delta_{n}^{i}-J_{n}^{i+1}\right|\\ \Delta_{n}^{i}=\left[x_{i},x_{i+1},\ldots,x_{i+n};f\right],\quad i=1,2,\ldots,m-n\\ x_{1}<x_{2}<\cdots<x_{m}\end{gathered}

when varying the points $x_{1},x_{2},\ldots,x_{m}$ and their number in all
possible ways over the domain of the function. We will say that the function has a bounded nth variation if its total variation of order $n$ is finished. We then study the functions whose $(n+1)$ The divided difference does not change sign. We will say that such a function is of order $n$ . For $n=0$ we have monotonic functions and for $n=1$ ordinary convex (or concave) functions. We point out the main properties of these functions and show their relationship with bounded divided nth difference functions and bounded nth variation functions.

If the function $f(x)$ is at ( $n+1$ jth difference divided by bounds we can obviously determine $\lambda$ such as the function $f(x)+\lambda x^{n+1}$ either of order $n$ We also show that a function with bounded nth variation is the difference of two functions of order $n$ .

We also study the differentiation of the functions defined previously, after having completed some of Stieltjes' research on the nth derivative of a function. We examine the limitation of the derivative of a function of order $n$ defined within an interval. We thus find that the order functions $n$ behave much like polynomials of degree $n$ at least within a suitably chosen inner interval

In the second part we try to extend to functions of two variables the results obtained for functions of a single variable.

The difference divided by order $(m,n)$ of $f(x,y)$ for the $k==(m+1)(n+1)$ points $M_{i}\left(x_{i},y_{i}\right),i=1,2,\ldots,k$ is equal to the quotient

\left[\mathrm{M}_{1},\mathrm{M}_{2},\ldots,\mathrm{M}_{k};f\right]_{m,n}=\frac{\mathrm{U}_{m,n}\left(\mathrm{M}_{1},\mathrm{M}_{2},\ldots,\mathrm{M}_{k};f\right)}{\mathrm{V}_{m,n}\left(\mathrm{M}_{1},\mathrm{M}_{2},\ldots,\mathrm{M}_{k}\right)}

Or $\mathrm{U}_{m,n}\left(\mathrm{M}_{1},\mathrm{M}_{2},\ldots,\mathrm{M}_{k};f\right)$ is the determinant whose general line is $1x_{i}x_{i}^{2}\ldots\ldots x_{i}^{m}y_{i}x_{i}y_{i}\ldots x_{i}^{m}y_{i}\ldots y_{i}^{n}x_{i}y_{i}^{n}\ldots x_{i}^{m-1}y_{i}^{n}f\left(x_{i},y^{i}\right)$ And $\mathrm{V}_{m,n}\left(\mathrm{M}_{1},\mathrm{M}_{2},\ldots,\mathrm{M}_{k}\right)=\mathrm{U}_{m,n}\left(\mathrm{M}_{1},\mathrm{M}_{2},\ldots,\mathrm{M}_{k};x^{m}y^{n}\right)$ We assume, of course, that the points $\mathrm{M}_{i}$ are such that the determinant $\mathrm{V}_{m,n}$ be different from zero.

We study these divided differences and show that a complete analogy can be established between the case of one and the case of two variables.

In the last Chapter we give a generalization of convex functions and doubly convex functions (See P. Montel, Journal de Math. 9th series, t. 7 (1928), p. 29-60) of two variables.

We are pleased to express here our deep gratitude to MP Montel who greatly encouraged us and whose valuable advice was very useful in the writing of this work.

Part one.

ON SOME PROPERTIES OF HIGHER-ORDER CONVEX FUNCTIONS OF A REAL VARIABLE

CHAPTER 1.

-ON THE DIVIDED DIFFERENCES OF THE FUNCTIONS OF A REAL VARIABLE.

§ 1. - Bounded divided difference functions and functions

with horned variation.

1.

We consider functions $f(x)$ defined, uniform and real of the real variable $x$ on a linear and bounded set E. To each point of E corresponds a finite and well-defined value for $f(x)$ We designate by a the left end and by $b$ the right end of set E. The points $a$ And $b$ are determined regardless of E. We denote by $\mathrm{E}^{\prime},\mathrm{E}^{\prime\prime},\ldots$ , the successive derived sets of k. We say that a set $\mathrm{E}_{1}$ is completely inside E if all its points belong to E and if its endpoints $a_{1},b_{1}$ are intercursed within the interval ( $a,b$ ), ( $a<a_{1}\leq b_{1}<b$ ).

We say that a sequence of points on the axis of the variable $x$ is - ordered, or that these points are ordered if their abscissas, referred to a fixed origin, are arranged in non-decreasing order. We also assume, unless otherwise stated, that all the points in such a sequence are distinct.

We call the polynomial of smallest degree l (Lagrange-Hermite polynomial)

\mathrm{P}\left(\alpha_{1}\right)=\mathrm{P}\left(\alpha_{1},\alpha_{2},\ldots\alpha_{k};f\mid x\right)

satisfying the conditions (1): (accents denote derivations)
( ² ) Hermite generalized Lagrange's polynomials in his memoir "On Lagrange's interpolation formula". Journal für die Reine und Bangew. Math t. 84 (1878) p. 70.

\begin{array}[]{ccc}\mathrm{P}\left(\alpha_{1}\right)=f\left(\alpha_{1}\right)&\mathrm{P}\left(\alpha_{p+1}\right)=f\left(\alpha_{p+1}\right)&\mathrm{P}\left(\alpha_{p+q+1}\right)=f^{\prime}\left(\alpha_{p+q+1}\right)\ldots\\ \mathrm{P}^{\prime}\left(\alpha_{2}\right)=f^{\prime}\left(\alpha_{2}\right)&\mathrm{P}^{\prime}\left(\alpha_{p+2}\right)=f^{\prime}\left(\alpha_{p+2}\right)&\mathrm{P}^{\prime}\left(\alpha_{p+q+2}\right)=f^{\prime}\left(\alpha_{p+q+2}\right)\ldots\\ \mathrm{P}(p-1)\left(\alpha_{p}\right)=f^{(p-1)}\left(\alpha_{p}\right)&\mathrm{P}^{(q-1)}\left(\alpha_{p+q}\right)=f^{(q-1)}\left(\alpha_{p+q}\right)&\mathrm{P}^{(r-1)}\left(\alpha_{p+q+r}\right)=f^{(r-1)}\left(\alpha_{p+q+r}\right)\\ \alpha_{1}=\alpha_{2}=\ldots=\alpha_{p},&\alpha_{p+1}=\alpha_{p+2}=\ldots=\alpha_{p+q},&\alpha_{p+q+1}=\alpha_{p+q+2}=\ldots=\alpha_{q+p+r},\ldots\\ p+q+r+\cdots=k.\end{array}

We know this polynomial is unique.
Finally, following M. Nörlund ^(2), we call it the divided difference of order: $\boldsymbol{k}$ of the function $f(x)$ for distinct points $\alpha_{1},\alpha_{2},\ldots\alpha_{k+1}$ the expression defined by the recurrence relation

\left[\alpha_{1},\alpha_{2},\ldots,\alpha_{k+};f\right]=\frac{\left[\alpha_{2},\alpha_{3},\ldots,\alpha_{k+1};f\right]-\left[\alpha_{1},\alpha_{2},\ldots,\alpha_{k};f\right]}{\left[\alpha_{k};f\right]=f(\alpha)}

(1)

The quantity (l) is symmetric with respect to the points $\alpha_{i}$ and can be expressed in the form of a quotient

\left[\alpha_{1},\alpha_{2},\ldots,\alpha_{k+1};f\right]=\frac{\mathrm{U}\left(\alpha_{1},\alpha_{2},\ldots,\alpha_{k+1};f\right)_{i}}{\mathrm{\penalty 10000\ V}\left(\alpha_{1},\alpha_{2},\ldots,\alpha_{k+1}\right)_{j}}

\mathrm{U}\left(\alpha_{1},\alpha_{2},\ldots,\alpha_{k+1};f\right)=\left|\begin{array}[]{cccccc}1&\alpha_{1}&\alpha_{1}^{2}&\ldots&\alpha_{1}^{k-1}&f\left(\alpha_{1}\right)\\ 1&\alpha_{2}&\alpha_{2}^{2}&\ldots&\ldots&\alpha_{2}^{k-1}\\ \ldots&\ldots&\ldots&\ldots&\ldots&\ldots\\ \ldots&\ldots&\ldots&\ldots&\ldots&\ldots\\ 1&\alpha_{k+1}&\alpha_{k+1}^{2}&\ldots&\alpha_{k+1}^{k-1}&\ldots\\ 1&\ldots&\ldots&\ldots\end{array}\right|

And

\mathrm{V}\left(\alpha_{1},\alpha_{2},\ldots,\alpha_{k+1}\right)=\mathrm{U}\left(\alpha_{1},\alpha_{2},\ldots,\alpha_{k+1};x^{k}\right)

is the Van der Monde determinant of quantities $\alpha_{i}$ From formula
(1) we can deduce others which we will point out as they are used. Note here that
(2)

f\left(\alpha_{k+1}\right)-\mathrm{P}\left(\alpha_{1},\alpha_{2},\ldots,\alpha_{k};\left.f\right|^{\frac{1}{\alpha}}\alpha_{k+1}\right)=

$=\frac{\mathrm{U}\left(\alpha_{1},\alpha_{2},\ldots,\alpha_{k+1};f\right)}{\mathrm{V}\left(\alpha_{1},\alpha_{2},\ldots,\alpha_{k}\right)}=\frac{\mathrm{V}\left(\alpha_{1},\alpha_{2},\ldots,\alpha_{k+1}\right)}{\mathrm{V}\left(\alpha_{1},\alpha_{2},\ldots,\alpha_{k}\right)}\cdot\left\lfloor\alpha_{1},\alpha_{2},\ldots,\alpha_{k+r};f\right]$
It follows that if $\left[\alpha_{1},\alpha_{2},\ldots,\alpha_{k+1};f\right]=0,f(x)$ takes on E the same values as a polynomial. We then say that $f(x)$ is a polynomial function.
2. Consider the divided differences

\left[x_{1},x_{2},\ldots,x_{n+1};f\right]

⁽²⁾ NE Nöblund “Lessons on Interpolation Series.” p. 2.
on all groups of $n+1$ distinct points of E. If E contains fewer than $n+1$ points we can indifferently assume that the nth divided difference does not exist or that it is identically zero.

Let's ask

\varlimsup_{(\operatorname{sur}E)}\cdot\left|\left[x_{1},x_{2},\ldots,x_{n+1};f\right]\right|=\Delta_{n}[f;E].

This number will be called the nth bound of $f(x)$ on $\mathrm{E}\Delta_{n}[f;\mathrm{E}]$ can also be referred to as $\Delta_{n}[f]$ or even $\Delta_{n}$ when there is no ambiguity and by $\Delta_{n}\stackrel{{\scriptstyle b}}\mid$ when it is a range ( $a,b$ ).

We say that the function is bounded divided nth difference on E if $\Delta_{n}$ It's over.

The case $n=0$ is that of bounded functions; $n=1$ that of functions satisfying an ordinary Lipschitz condition.
3. Consider $m$ ordered points

x_{1},x_{2},\ldots,x_{m}

(3)

—

and be

	$\displaystyle\dot{\Delta}_{k}^{i}=\left[x_{i},x_{i+1},\ldots,x_{i+k};f\right]$		(4)
	$\displaystyle\hat{\imath}=1,2,\ldots,m-k,\quad k=0,1,2,\ldots,m-1$		(i)

the divided differences of a function defined at these points.
The sum

v_{n}=\sum_{i=1}^{m-n-1}\left|\Delta_{n}^{i+1}-\Delta_{n}^{i}\right|

(5)

is the nth variation of $f(x)$ on points (3).
That is $f(x)$ defined on a set E. The variations $v_{n}$ on all ordered sequences of E have an upper limit $\mathrm{V}_{n}[f;\mathrm{E}]$ We refer to this number as $\mathrm{V}_{n}[f],\mathrm{V}_{n}$ Or $\mathrm{V}_{n}[f]$ and we call it the nth total variation of $f(x)$ on E.

We will say that the function is bounded on E with nth variation if $V_{n}$ It's over.

The case $n=0$ is that of ordinary bounded variation functions of Jordan ${}^{(3)};n=1$ has already been implicitly considered by Mr. De la
( ³ ) For the study of these functions see H. Lebesgue “Lessons on integration … etc." $2^{\text{ème }}$ ed. (1928) p. 96; or even L. Tonelli „Fondamenti di Calcolo della variazioni” t. I, p. 40.

Vallée Poussin ( ⁴ ) and studied in a general way by M.A. Winternitz $\left({}^{5}\right)$ .

S. 2. - Properties of functions whose nth difference divided

is bounded.
4. Let $\alpha_{1},\alpha_{2},\ldots,\alpha_{j};\beta_{1},\beta_{2},\ldots,\beta_{j};\alpha_{j+1}=\beta_{j+1},\alpha_{j+2}=\beta_{j+2},\ldots,{}_{2}\alpha_{k}=\beta_{k},k+j$ distinct points ( $1\leq j\leq k$ ). We have, according to formula (1)
$\left(\alpha_{i}-\beta_{i}\right)\left[\alpha_{1},\alpha_{2},\ldots,\alpha_{i},\beta_{i},\beta_{i+1},\ldots,\beta_{k};f\right]=$

=\left[\alpha_{1},\alpha_{2},\ldots,\alpha_{i},\beta_{i+1},\ldots,\beta_{kj};f\right]-\left[\alpha_{1},\alpha_{2},\ldots,\alpha_{i-1},\beta_{i},\ldots,\beta_{k};f\right]_{j}

Doing $i=1,2,\ldots,k$ Adding term by term and removing identically null terms, we deduce the following formula:
(6) $\left[\alpha_{1},\alpha_{2},\ldots,\alpha_{k};f\right]-\left[\beta_{1},\beta_{2},\ldots,\beta_{k};f\right]=$

=\sum_{i=1}^{j}\left(\alpha_{i}-\beta_{i}\right)\left[\alpha_{1},\alpha_{2},\ldots,\alpha_{i},\beta_{i},\beta_{i+1},\ldots,\beta_{k};f\right]

(For $j=1$ we have formula (1) itself).
This formula allows us to write
(7) $\left.\left|\left[x_{1},x_{2},\ldots,x_{n};f\right]\right|\leq\left|\left[x_{1}^{\prime},x_{2}^{\prime},\ldots,x_{n}^{\prime};f\right]\right|+\left(\sum_{i=1}^{n}\left|x_{i}-x_{i}^{\prime}\right|\right)\right\}.A_{n}[f\mid]$

<\left|\left[x_{1}^{\prime},x_{2}^{\prime},\ldots,x_{n}^{\prime};f\right]\right|+n(b-a).\Delta_{n}[f]

therefore any function with a bounded divided difference nth is also ( $n-1$ )th difference divided bounded.

In particular, any bounded divided difference function with nth difference is bounded.

We can also see that the function has a bounded number of derivatives if $n>1$ ; it is therefore also continuous in this case.

Continuity also results from the following formula:
(8) $\left|\left[x_{1},x_{2},\ldots,x_{n+1};f\right]-\left[x_{1}^{\prime},x_{2}^{\prime},\ldots,x_{n+1}^{\prime};f\right]\right|<2n\Delta_{n}$ . §

\delta_{i}=\max.\left|\frac{x_{i}-x^{\prime}}{\delta_{i}}\right|\leq 1\left({}^{6}\right)

(4) Ch. de la Vallée Poussin "Note on the approximation by a polynomial of a function whose derivative has bounded variation" ${}^{\text{" }}$ Bull. Aead. Belgium1908 p. 403.
( ⁵ ) A. Winternitz "Uber eine Klasse von linearen Funktional Ungleichungen und über konvexe Funktionale ^4. Berichte kön, sächsischen Gesellschs der Wissensch. zu Leipzig t. 69 (1917) p. 349.
(6) By the max notation $\left(a_{1},a_{2},\ldots\right.$ ) Or $\max_{i=1,2,\ldots}\left(a_{i}\right)$ we designate the largest of the numbers $a_{1},a_{2},\ldots$ . Similar notation for the smallest of these numbers.
$\delta=$ length of the smallest interval containing
the points $x_{1},x_{2},\ldots,x,x_{i}^{\prime},x_{i+1}^{\prime},\ldots,x_{n+1}^{\prime}$

i=1,2,\ldots,n+1

Either $x^{\prime}$ a point of $\mathrm{E}^{\prime}$ not belonging to E. If, regardless of how the point $x$ E tends towards $x^{\prime},f(x)$ Since the function tends towards the same finite and well-defined limit, we can still say that the function is continuous at the point $x^{\prime}$ by taking $f(\dot{x})$ equal to this limit.

There always exists a countable subset $\mathrm{E}^{*}$ of E such that $\mathrm{E}-\mathrm{E}^{*}$ belongs to $\mathrm{E}^{*\prime}$ and that the function continues $f(x)$ be completely determined by its values on $\mathrm{E}^{*}$ .

Formula (1) also allows us to establish the following:

		$\displaystyle\left(\alpha_{k+1}-\alpha_{1}\right)\left[\alpha_{1},\alpha_{2},\ldots,\alpha_{i-1},\alpha_{i+1},\ldots,\alpha_{k+1};f\right]=$		(9)
	$\displaystyle=$	$\displaystyle\left(\alpha_{i}-\alpha_{1}\right)\left[\alpha_{1},\alpha_{2},\ldots,\alpha_{k};f\right]+\left(\alpha_{k+1}-\alpha_{i}\right)\left[\alpha_{2},\alpha_{3},\ldots,\alpha_{k+1};f\right].$

If the following $\alpha_{1},\alpha_{2},\ldots,\alpha_{k+1}$ is ordered, we can see that the $d$ divided frene $\left[\alpha_{1},\alpha_{2},\ldots,\alpha_{l-1},\alpha_{i+1},\ldots,\alpha_{k+1};f\right]$ is included between the divided differences $\left[\alpha_{1},\alpha_{2},\ldots,\alpha_{k};f\right],\left[\alpha_{2},\alpha_{3},\ldots,\alpha_{k+1};f\right]$ .

Either $\alpha_{1}^{\prime},\alpha_{2}^{\prime},\ldots,\alpha_{k+1}^{\prime}$ a partial sequence extracted from the ordered sequence $\alpha_{1},\alpha_{2},\ldots,\alpha_{m}$ and such that $\alpha_{1}^{\prime}=\alpha_{1},\alpha_{k+1}^{\prime}=\alpha_{m}$ .

By repeatedly applying formula (9) we obtain

\left[\alpha_{1}^{\prime},\alpha_{2}^{\prime},\ldots,\alpha_{k+1}^{\prime};f\right]=\sum_{i=1}^{m-k}\mathrm{\penalty 10000\ A}_{l}\left[\alpha_{i},\alpha_{i+1},\ldots,\alpha_{i+k};f\right]\left({}^{7}\right)

(10)

where the $\mathrm{A}_{i}$ are positive, independent of the function $f(x)$ , and have a sum equal to 1.

It follows that

\begin{array}[]{r}\min_{i=1,2,\ldots,m-k}\left(\left[\alpha_{i},\alpha_{i+1},\ldots,\alpha_{i+k};f\right]\right)\leq\left[\alpha_{1}^{\prime},\alpha_{2}^{\prime},\ldots,\alpha_{k+1}^{\prime};f\right]\leq\\ \leq\max_{i=1,2,\ldots,m-k}\left(\left[\alpha_{i},\alpha_{i+1},\ldots,\alpha_{i+k};f\right]\right)\end{array}

( ⁷ ) It can be noted in general that if the sum

\sum_{i=1}^{m-k}\mathrm{\penalty 10000\ A}_{l}\left[\alpha_{l},\alpha_{i+1},\ldots,\alpha_{i+k};f\right]\quad\left(\mathrm{A}_{l}\text{ indépendanls de }f(x)\right)

depends explicitly only on $f\left(\alpha_{1}^{\prime}\right)_{1},f\left(\alpha_{2}^{\prime}\right),\ldots,f\left(\alpha_{p}^{\prime}\right)_{1}\alpha_{1}^{\prime},\alpha_{2}^{\prime},\ldots,\alpha_{p}^{\prime}$ being a partial sequence of $\alpha_{1},\alpha_{2},\ldots,\alpha_{m}$ it is necessarily of the form

\sum_{i=1}^{p-k}\mathrm{\penalty 10000\ A}_{i}^{\prime}\left[\alpha_{i}^{\prime},\alpha^{\prime}+1,\ldots,\alpha^{\prime}{}_{i+k};f\right]\quad\left(\mathrm{A}^{\prime}{}_{i}\text{ indépendants de }f(x)\right).

A special case of formula (10) was used by MA Marchaud in his Thesis "On the derivatives and differences of functions of real variables" (Paris 1927) p. 32.
and

\left|\left[\alpha_{1}^{\prime},\alpha_{2}^{\prime},\ldots,\alpha_{k+1}^{\prime};f\right]\right|\leq\max_{i=1,2,\ldots,m-k}\left(\left|\left\{\alpha_{l},\alpha_{i+1},\ldots,\alpha_{i+k};f\right]\right|\right).

Suppose that E is an interval and are $x_{1},x_{2},\ldots,x_{n+1}x_{1}^{\prime},x_{2}^{\prime},\ldots,x_{n+1}^{\prime},2n+2$ points of this interval.

Let's suppose that $n>0$ and let's write

\xi_{i}=\frac{x_{1}+\lambda x_{i}^{\prime}}{1+\lambda},\quad i=1,2,\ldots,n+1.

Formula (8) shows that $\left|\xi_{1},\xi_{2},\ldots,\xi_{n+1};f\right|$ is a continuous function of $\lambda$ For $\lambda\geq 0$ , equal for $\lambda=0$ has $\left[x_{1},x_{2},\ldots,x_{n+1};f\right]$ and for $\lambda=+\infty$ has $\left[x^{\prime}{}_{1},x^{\prime}{}_{2},\ldots,x^{\prime}{}_{n+1};f\right]$ .

We can therefore deduce the following property:
If $\mathbf{E}$ is an interval $\epsilon t$ if $\left[x_{1},x_{2},\ldots,x_{n+1};f\right]=\mathbf{A}$ ,
$\left[x_{1}^{\prime},x_{2}^{\prime},\ldots,x_{n+1}^{\prime};f\right]=\mathrm{B}$ , there exists in every interval containing all the points $x_{i},x_{i}^{\prime}$ a divided difference taking any value between $A$ ei $B$ .

The property is not true for $n=0$ because in this case we always have $\delta=1$ in formula (8).

In particular, if the nth divided difference remains in modulus greater than a positive number, it retains a constant sign.

Either $x_{1}^{*},x_{2}^{*},\ldots,x_{m}^{*}$ an ordered sequence such as $x_{1},x_{2},\ldots,x_{n+1}$ in itself a partial sequence ( $x_{1}=x_{1}^{*},x_{n+1}=x^{*}{}_{m}$ ) and such that

\max_{i=1,2,\ldots,m-1}\left(\left|x_{i+1}^{*}-x_{i}^{*}\right|\right)<\frac{\varepsilon}{n+1}

$\varepsilon$ being a positive number.
Let's consider the divided differences

\text{ (11) }\quad d_{i}=\left[x_{i}^{*},x_{i+1}^{*},\ldots,x_{i+n}^{*};f\right]\quad i=-1,2,\ldots,m-n\text{. }

Now suppose that $\left[x_{1},x_{2},\ldots,x_{n+1};f\right]=0$ and let's apply formula (10). We then see that there must be at least one index $i$ for which $d_{i}d_{i+1}\leq 0$ From this inequality and the property previously demonstrated, we deduce that

If $\left[x_{1},x_{2},\ldots,x_{n+1};f\right]=0$ there exists, in the smallest interval containing the points $x_{i}$ , an interval of any length we want where there is at least one zero divided difference.

By applying the property to the function $f-\mathrm{A}x^{n}$ and looking more closely at the demonstration, we see that if $\left[x_{1},x_{2},\ldots,x_{n+1};f\right]=\mathrm{A}$ , there is a point $x$ in the smallest interval containing the points $x_{i}$ such that in every interval $\mathrm{I}(x;\eta)$ middle $x$ and length $\eta$ there exists at least one divided difference taking the value $A_{1}$

We can also see that, from formula (6), we can deduce that if E is an interval and if

\begin{gathered}{\left[x_{1},x_{2},\ldots,x_{n};f\right]=0,\quad\left[x_{1}^{\prime},x_{2}^{\prime},\ldots,x_{n}^{\prime};f\right]=0}\\ x_{1},x_{1}^{\prime},x_{2},x_{2}^{\prime},\ldots,x_{n},x_{n}^{\prime}\quad\text{ ordonnés }\end{gathered}

we can find in the interval ( $x_{1},x_{n}^{\prime}$ ) at least one nth divided difference equals zero.
5. It follows from the definition that the nth bound on a subset is at most equal to $\Delta_{n}$ .

Either $c$ a point of $E$ And $E_{1},E_{2}$ the parts of $E$ respectively included in the closed intervals $(a,c),(c,b)$ We will show that if the set E is dense in the interval ( $a,b$ ) the nth iborne of $f(x)$ is equal to $\Delta_{n}[f;E]$ on at least one of the sets $\mathbb{E}_{1},\mathrm{E}_{2}$ .

For $n=0$ The property is obvious, regardless of E.
Let us therefore assume $n>0$ and consider a sequence of positive numbers $\varepsilon_{1},\varepsilon_{2},\ldots,\varepsilon_{m},\ldots$ tending towards zero with $\frac{1}{m}$ By definition, there exists a divided difference such that

\left.\mid x_{1},x_{2},\ldots,x_{n+1};f\right]\left\lvert\,>\Delta_{n}-\frac{\varepsilon_{m}}{2}>\Delta_{n}-\varepsilon_{m}\right.

If the points $x_{i}$ are on the same side of the point $c$ we take this difference divided and we designate it by $\Delta_{n}\left(\varepsilon_{m}\right)$ Otherwise, by applying formula (9) if necessary, we can assume that $c$ intervenes in the divided difference under consideration.

So be it

\left|\left[x_{1},x_{2},\ldots,x_{i},c,x_{i+1},\ldots,x_{n};f\right]\right|>\Delta_{n}-\frac{\varepsilon_{m}}{2}

(12)

The sequel $x_{1},x_{2},\ldots,x_{l},c,x_{i+1},\ldots,x_{n}$ being ordered. Let's apply formula (9) by inserting a new point between $c$ , $x_{i+1}$ [which is always possible since E is assumed to be dense in the interval ( $a,b$ )] and take the one of the divided differences that satisfies inequality (12). By repeating this procedure, two cases can arise:
10. Or, there always remains $i$ points to the left of $c$ And so there are divided differences. $\left[x_{1},x_{2},\ldots,x_{i},c,x_{i+1}^{\prime},\ldots,x_{n}^{\prime};f\right]$ verifying (12), the points $x_{i+1}^{\prime},\ldots,x_{n}^{\prime}$ being as close as we want to c, we can then find an ordered sequence. $x_{1},x_{2},\ldots,x_{i},x_{i}^{\prime\prime}{}_{1},\ldots,x_{n}^{\prime\prime}$ , c such that formula (8) gives us
$\|\left[x_{1},x_{2},\ldots,x_{i},x^{\prime\prime}{}_{i+1},\ldots,x^{\prime\prime}{}_{n},c;f\right]\left|>\left|\left[r_{1},x_{2},\ldots x_{i},c,x^{\prime}{}_{i+1},\ldots,x^{\prime}{}_{n};f\right]\right|-\frac{\varepsilon_{m}}{2}\right.$
and then

\left|\left[x_{1},x_{2},\ldots,x_{i},x_{i+1}^{\prime\prime},\ldots,x_{n}^{\prime\prime},c;f\right]\right|>\Delta_{n}-\varepsilon_{m}.

We take this difference divided for $\Delta_{n}\left(\varepsilon_{m}\right)$ .
$2^{0}$ Or, at some point, there is only $i-1$ points on the left: of $c$ and we are brought back to the case $1^{0}$ .

Finally, it therefore still exists in $\mathrm{E}_{1}$ Or $\mathrm{E}_{2}$ a difference divided $\Delta_{n}\left(\varepsilon_{m}\right)$ verifying inequalities

The sequel

\Delta_{n}\geq\left|\Delta_{n}\left(\varepsilon_{m}\right)\right|>\Delta_{n}-\varepsilon_{m}

\Delta_{n}\left(\varepsilon_{1}\right),\Delta_{n}\left(\varepsilon_{2}\right),\ldots,\Delta_{n}\left(\varepsilon_{m}\right),\ldots

therefore has a limit $\Delta_{-}$ Now, there is certainly an infinite partial sequence located entirely in $\mathfrak{E}_{1}$ Or $\mathfrak{E}_{2}$ And this sequel obviously has the same limitation. $\Delta_{n}$ , which proves ownership.

We can easily deduce that
the nth bound of a function defined and with nth difference divided by bounds on a dense set in an interval ( $a,b$ ) is the same as on a subset contained within a subinterval of ( $a,b$ ), of any length you want.

The previous demonstration also shows us that if E is dense in an interval, the upper limit of $\left\{x_{1},x_{2},\ldots,x_{n+1};f\right\}\mid$ is equal to its greatest limit.

Formula (9) further shows us that if $\left|\left[x_{1},x_{2},\ldots,x_{n+1};f\right]\right|=\Delta_{n}$ the nth difference divided is constantly equal to $\left\{x_{1},x_{2},\ldots,x_{n+1};f\right\}$ in the smallest interval containing the points $x_{i}$ .

The study of functions whose divided nth difference is a constant A is equivalent to that of functions with zero nth difference, since $f-\mathrm{A}x^{n}$ is such a function. Therefore, it is a polynomial function. To be more precise, we will say that it is a polynomial function of order $n-1$ It takes on E the values of a polynomial of degree $n-1$ .

If the terminal $\Delta_{n}$ is reached by $\left[x_{1},x_{2},\ldots,x_{n+1};f\right]$ the function is polynomial of order $n$ on the part of E contained within the smallest interval containing the points $x_{i}$ . If $\Delta_{n}>0$ , the degenerate of the polynomial: is indeed equal to $n$ 6.
Between two terminals $\Delta_{n},\Delta_{m}$ There is generally no relationship. Let's return to formula (7). Relation (1) shows us that
(13)

\left|\left[x_{1}^{\prime},x_{2}^{\prime},\ldots,x_{n}^{\prime};f\right]\right|\leq A_{k}.\Delta_{k},k<n-1

Or $\mathbf{A}_{\boldsymbol{k}}$ depends on the points $x^{\prime}{}_{\boldsymbol{i}}$ The minimum A of $\mathrm{A}_{k}$ depends only on the set E. We then have
(14)

\Delta_{n-1}\leq\mathrm{A}\cdot\Delta_{k}+\mathrm{B}\cdot\Delta_{n},k<n-1

Or $\mathrm{A},\mathrm{B}$ depend only on the set E.
Generally speaking, between three bounds $\Delta_{p},\Delta_{q},\Delta_{r},p<q<r$ ir: $y$ always has a relationship of the form $\Delta_{q}\leq\mathrm{A}.\Delta_{p}+\mathrm{B}.\Delta_{r}$ where A and B depend only on the set E.

In particular, suppose that E is a closed interval $(a,b)$ We can write

	$\displaystyle\mid\left[x_{1}^{\prime},x_{2}^{\prime},\ldots,x_{n}^{\prime};fj\mid\right.$	$\displaystyle\leq\frac{2}{\left\|x_{1}^{\prime}-x_{n}^{\prime}\right\|}\cdot\Delta_{n-2}$
	$\displaystyle\min\cdot\frac{2}{\left\|x_{1}^{\prime}-x_{n}^{\prime}\right\|}$	$\displaystyle=\frac{2}{b-a}$

The results of No. 5 show us that in order to minimize the coefficient of $J_{n}$ in formula (7) it is permissible to take

x_{2}^{\prime}=x_{3}^{\prime}=\cdots=x_{n-1}^{\prime}=x_{1}=x_{2}=\cdots=x_{n}

We then obtain formula
(15)

\Delta_{n-1}\leq\frac{2}{a-b}\leftharpoonup_{n-2}+(b-a)\Delta_{n}

\begin{gathered}\left|\left[x_{1}^{\prime},x_{2}^{\prime},\ldots,x_{n}^{\prime};f\right]\right|\leq\frac{2\left(x_{n}^{\prime}+x_{n-1}^{\prime}-x_{1}^{\prime}-x_{2}^{\prime}\right)}{\left(x_{n}^{\prime}-x_{1}^{\prime}\right)\left(x_{n-1}^{\prime}-x_{1}^{\prime}\right)\left(x_{n}^{\prime}-x_{2}^{\prime}\right)}\Delta_{n-3}\\ n>3,x_{1}^{\prime}<x_{2}^{\prime}<x_{n-1}^{\prime}<x_{n}^{\prime}\end{gathered}

\Delta_{n-1}\leq\frac{4}{(b-a)^{2}}\Delta_{n-3}+2(b-a)\Delta_{n}\quad(n>3)

Let's take another one

The minimum of the coefficient of $\Delta_{0}$ is equal to ( ⁸ )

\frac{2^{2n-3}}{(b-a)^{n-1}}

(8) The maximum of the inverse of this quantity is indeed equal to the _maximum of the polynomial $x^{n-1}+\ldots$ deviating as little as possible from zero in the interval ( $a,b$ ). See Ch. de la Vallée Potjssin, "Lessons on the Approximation of Functions of a Real Variable," Chapter VI. The polynomial in question is:

\frac{(b-a)^{n-1}}{2^{2n-8}}\cos\left[(n-1)\arccos\frac{2x-a-b}{b-a}\right]

See: S. Bernstein "Lectures on Extreme Properties . . . etc." p. 6.
with

x_{i}^{\prime}=\frac{b+a}{2}-\frac{b-a}{2}\cos\frac{i-1}{n-1}\pi,\quad i=1,2,\ldots,n.

We then have

\sum_{i=1}^{n}\left|x_{i}-x_{i}^{\prime}\right|\leq\sum_{i=1}^{n}\left|x_{i}^{\prime}-a\right|=\frac{b-a}{2}\left(n-\sum_{i=1}^{n}\cos\frac{i-1}{n-1}\pi\right)=\frac{n}{2}(b-a)

-hence the relation
'(17)

\Delta_{n-1}\leq\frac{2^{2n-8}}{(b-a)^{n-1}}\Delta_{0}+\frac{n}{2}(b-a)\Delta_{n}

In relations (15), (16), (17) we can obviously replace $-b-a$ by a smaller number.

The preceding inequalities are those of Mr. Hadamard $\left({}^{9}\right)$ when we assume the existence of the nth derivative. We obtained them by a very simple method.
7. If $f$ And $\varphi$ are at nth difference divided by bounded, $f+\varphi$ , ef where $c$ is a constant are also at nth difference divided bounded.

The product of two functions with a bounded divided nth difference is still a bounded divided nth difference. This follows from the formula $\left({}^{10}\right)$

	$\displaystyle{\left[\alpha_{i},\alpha_{2},\ldots,\alpha_{k+1};f.\varphi\right]=\sum_{t=0}^{k}\left[\alpha_{1},\alpha_{2},\ldots,\alpha_{k-i+1};f\right].}$		(18)
	$\displaystyle{\left[\alpha_{k-i+1},\alpha_{k-i+2},\ldots,\alpha_{k+1};\varphi\right]}$

which can be easily verified by induction using (1).
More generally, if $F$ And $f$ are at nth difference divided bounded , $\mathbf{F}(f)$ It is too. This will result from a formula giving the difference - divided by a function of a function.

Let's state this.

f_{i}=f\left(\alpha_{i}\right),\quad i=1,2,\ldots,k+1

We obviously have a relationship of the form

	$\displaystyle{\left[\alpha_{1},\alpha_{2},\ldots,\alpha_{k+1};\mathrm{F}(f)\right]=\sum_{i=1}^{k}\left[f_{i},f_{i+1},\ldots,f_{k+1};\mathrm{F}\right].}$		(19)
	$\displaystyle.\mathrm{A}_{1}^{(k)}\left(\alpha_{1},\alpha_{2},\ldots,\alpha_{k+1}\right)$

(9) Voir par ex. T. Carleman "Les fonctions quasi-analytiques" (Paris, 1926) Chap. II.
( ¹⁰ ) C’est l’analogun en termes finis de la formule de Leibnitz. Pour des points équidistants elle a été signalée par M. E. Jacobsthal „Mittelwertbildung und Reihentransformation" Math. Zeitschr. t. 6 (1920) p. 100.
les $\mathrm{A}_{i}^{(k)}\left(\alpha_{1},\alpha_{2},\ldots,\alpha_{k+1}\right)$ ne dépendant que de la fonction $f(x)$ . Ces : quantités peuvent se calculer à l’aide des relations de récurrence

\begin{gathered}A_{i}^{(k)}\left(\alpha_{1},\alpha_{2},\ldots,\alpha_{k+1}\right)=\sum_{j=1}^{i}\left[\alpha_{1},\alpha_{2},\ldots,\alpha_{k+1};f\right]\cdot A_{i-j+1}^{(k+1)}\left(\alpha_{j},\alpha_{j+1},\ldots,\alpha_{k}\right)\\ A_{k}^{(k)}\left(\alpha_{1},\alpha_{2},\ldots,\alpha_{k+1}\right)=\left[\alpha_{1},\ldots,\alpha_{k+1};f\right)\end{gathered}

Si nous désignons par $d_{r}^{\prime},d_{r}^{\prime\prime},,\ldots,d_{r}^{(p)}$ des différences divisées : d’ordre $r$ de la fonction $f(x)$ sur des points $\alpha_{i}$ , convenablement choisis le coefficient $A_{i}^{(k)}\left(\alpha_{1},\alpha_{2},\ldots,\alpha_{k+1}\right)$ est de la forme

\sum d_{1}^{\prime}d_{1}^{\prime\prime}\ldots d_{1}^{(p)}d_{2}^{\prime}d_{2}^{\prime\prime}\ldots d_{2}^{(q)}\ldots d_{k}^{\prime}d_{k}^{\prime\prime}\ldots\dot{a}_{k}^{(t)}

(20)

avec

p+q+\cdots+t=k-i+1,\quad p+2q+\cdots+kl=k\left({}^{11}\right)

(21)

Par exemple si $f$ est à nème différence divisée bornée, $f^{k}$ l’est : aussi si $k\geq n\left({}^{12}\right)$ , ou bien si $k$ est un entier positif. Si $|f|>c>0$ . sur $\mathrm{E},f^{k}$ est à nème différence divisée bornée quel que soit $k$ . On en $\mathrm{en}^{\mathrm{a}}$ déduit que le quotient de deux fonctions à nème différence divisée bornée l’est aussi si le dénominateur reste en module plus grand qu’un nombre positif. Nous en déduisons aussi que le module d’une fonction à nème différence divisée bornée n’est pas en général à nème différence : divisée bornée si $n>1$ .

Considérons une famille de fonctions ( $f$ ) définies sur un même ensemble E. Désignons par $\Delta_{n}^{*}$ la limite supérieure des nèmes bornes des fonctions de cette famille. On voit tout d’abord que si $\Delta_{n}^{*}$ est fini. toute fonction limite de la famille est à nème différence divisée bornéeet sa borne ne dépasse pas $\Delta^{*}{}_{n}$ . Si $\Delta^{*}{}_{n}$ est fini, il ne résulte pas encoreque $\Delta_{0}^{*},\Delta_{1}^{*},\ldots,\Delta_{n-1}^{*}$ sont finis. Mais si $\Delta_{n}^{*},\Delta_{m}^{*}(m<n)$ sont finisil résulte des inégalités de M. Hadamard que $\Delta_{m+1}^{*},\Delta_{m+2}^{*},\ldots,\Delta_{n-1}^{*}$ sont aussi finis. Si les fonctions de la famille ne sont pas définies surle même ensemble des circonstances toutes différentes peuvent se présenter.

Considérons une suite d’ensembles finis $\mathrm{E}^{*}{}_{1},\mathrm{E}^{*}{}_{2},\ldots,\mathrm{E}_{m}^{*},\ldots$ chacun contenant le précédent et ayant pour limite $\mathrm{E}^{*}$ , évidemment dénom-. brable ; inversement, tout ensemble dénombrable peut s’obtenir de
(11) La somme (20) s’étend à toutes les solutions en nombres entiers et. positifs du système (21) et à chaque solution correspondent $\frac{(k-i+1)!}{p!q!\ldots t!}$ termes _… La formule (19) donne à la limite la dérivée kème d’une fonction de fonction
⁽¹²⁾ On considère bien entendu une branche réelle de la fonction $f^{k}$ .
cette manière. Soit une suite de fonctions $f_{1},f_{2},\ldots,f_{m},\ldots,f_{m}$ étant définie sur $\mathrm{E}_{m}^{*},m=1,2,\ldots$ , et telle que

\Delta_{0}\left[f_{m};\mathrm{E}_{m}^{*}\right]\leq\Delta_{0},\quad\Delta_{n}\left[f_{m};\mathrm{E}_{m}^{*}\right]\leq\Delta_{n},\quad m=1,2,\ldots

Il existe alors au moins une fonction limite $f(x)$ définie sur $\mathrm{E}^{*}$ wérifiant les inégalités

\Delta_{0}\left[f;\mathrm{E}^{*}\right]\leq\Delta_{0},\quad\Delta_{n}\left[f;\mathrm{E}^{*}\right]\leq\Delta_{n}

La démostration est immédiate $\left({}^{13}\right)$ .

§ 3. Propriétés des fonctions à nème variation bornée.

8.

La formule (6) donne
(22) $\left|\left[x_{1},x_{2},\ldots,x_{n+1};f\right]\right|\leq\leq\left|\left[x_{1}{}^{\prime},x_{2}{}^{\prime},\ldots,x_{n+1}^{\prime};f\right]\right|+(n+1)V_{n}$ donc, toute fonction à nème variation bornée est à nème différence divisée bornée.

La réciproque n’est pas vraie.
De (1) et de (5) nous déduisons $\mathrm{V}_{n}\leq(n+1)(b-a)\Delta_{n!1}$ , donc toute tonciion à ( $n+1$ )ème différence divisée bornée est à nème variation bornée.

La réciproque n’est pas vraie ( ⁽¹⁴⁾.
Il èn résulte que toute fonction à nème variation bornée est aussi à $(n-1)$ ème variation bornée. En particulier une telle fonction est toujours bornée.

Posons $v_{n}=v_{n}\left(x_{1},x_{2},\ldots,x_{m}\right)$ en mettant en évidence les points (3). Soit $v_{n}^{(m)}$ la limite supérieure des $v_{n}\left(x_{1},x_{2},\ldots,x_{m}\right)$ lorsque les points varient sur E, leur nombre restant fixe. A tout $\varepsilon>0$ correspond donc au moins une suite (3) telle que

v_{n}\left(x_{1},x_{2},\ldots,x_{m}\right)>v_{n}^{(m)}-\varepsilon

On démontre facilement, à l’aide de la formule (9), que si on ajoute un nouveau point $x$ , compris par exemple entre $x_{i},x_{i+1}$ , on a

v_{n}\left(x_{1},x_{2},\ldots,x_{l},x_{,}x_{i+1},\ldots,x_{m}\right)\geq v_{n}\left(x_{1},x_{2},\ldots,x_{m}\right)

d’où

v_{n}^{(m+1)}>v_{n}^{(m)}-\varepsilon

⁽¹³⁾ La démonstration se fait par la méthode diagonale bien connue. Grâce aux travaux de M. Montel, c’est aujourd’hui une méthode courante dans ce genre de problèmes.
( ¹⁴ ) Il est facile de mettre en défaut la réciproque par des fonctions convenablement choisies et par des intégrations répétées.
donc

v_{n}^{(m+1)}\geq v_{n}^{(m)}

La quantité $v_{n}^{(m)}$ tend donc pour $m\rightarrow\infty$ vers une limite, qui est nécessairement égale à $\mathrm{V}_{n}$ . Si E contient $m$ points $\mathrm{V}_{n}=\mathrm{V}_{n}^{(m)}$ . Si E contient une infinité de points $V_{n}$ est aussi la plus grande des limites des $v_{n}$ .

Il est à peu près évident que, si E* est un sous-ensemble de E, on a $\mathrm{V}_{n}\left[f;\mathrm{E}^{*}\right]\leq\mathrm{V}_{n}[f;\mathrm{E}]$ .

Prenons un point $c$ appartenant à $\mathrm{E}+\mathrm{E}^{\prime}$ et désignons par $\mathrm{E}_{4},\mathrm{E}_{2}$ , des parties de E comprises dans les intervalles fermés $(a,c),(c,b)$ .

Il est facile de voir que

\mathrm{V}_{n}\left[f;\mathrm{E}_{1}\right]+\mathrm{V}_{n}\left[f;\mathrm{E}_{2}\right]\leq[f;\mathrm{E}].

Supposons maintenant que $c$ appartienne à E. Soit $v^{\prime}{}_{n}$ la variation sur les points (3) auxquels on ajoute le point $c$ et $v_{n}^{\prime\prime},v_{n}^{\prime\prime\prime}$ les variations sur les points de cette suite qui sont respectivement dans $\mathrm{E}_{1}$ et $\mathbb{E}_{2}$ . On peut prendre les points (3) de manière que

v_{n}^{(m)}-\varepsilon<v_{n}\leq v_{n}^{\prime}\leq v_{n}^{\prime\prime}+v_{n}^{\prime\prime\prime}

d’où

\mathrm{V}_{n}[f;\mathrm{E}]\leq\mathrm{V}_{n}\left[f;\mathrm{E}_{1}\right]+\mathrm{V}_{n}\left[f;\mathrm{E}_{2}\right].

Nous avons donc dans ce cas

\mathrm{V}_{n}[f;\mathrm{E}]=\mathrm{V}_{n}\left[f;\mathrm{E}_{1}\right]+\mathrm{V}_{n}\left[f;\mathrm{E}_{2}\right]

(23)

9.

Si $n>0$ , une fonction à nème variation bornée est continue.

Soit E* un sous ensemble dénombrable de E, tel que E-E* apparđienne tout entier au dérivé de $\mathrm{E}^{*}$ ( $\mathrm{E}^{*}$ peut coïncider avec E ).

A toute variation $v_{n}$ et à tout nombre $\varepsilon>0$ , on peut faire correspondre une variation $v^{*}{}_{n}$ sur $\mathrm{E}^{*}$ telle que $v_{n}<v^{*}{}_{n}+\varepsilon$ , donc : $\mathrm{V}_{n}[f;\mathrm{E}]\leq\mathrm{V}_{n}\left[f;\mathrm{E}^{*}\right]$ .

Mais on a aussi $\mathrm{V}_{n}[f;\mathrm{E}]\geq\mathrm{V}_{n}\left[f;\mathrm{E}^{*}\right]$ , donc :

\mathrm{V}_{n}[f;\mathrm{E}]=\mathrm{V}_{n}\left[f;\mathrm{E}^{*}\right]

Nous pouvons toujours trouver une suite d’ensemble $\mathrm{E}^{*}$ finis $\mathbb{E}^{*}{}_{1},\mathrm{E}^{*}{}_{2},\ldots,\mathrm{E}^{*}{}_{m},\ldots$ , chacun contenant le précédent, telle que la fonction soit complètement déterminée par ces valeurs sur la limite $\mathrm{E}^{*}$ de cette suite, et telle aussi qua

\lim_{m\rightarrow\infty}\mathrm{\penalty 10000\ V}_{n}\left[f;\mathrm{E}_{m}^{*}\right]=\mathrm{V}_{n}[f;\mathrm{E}]

Ces propriétés résultent de la continuité. Elles restent donc vraies pour $n=0$ si la fonction est continue. On voit aussi que dans ce cas (23) reste vraie même si $c$ est un point de $E^{\prime}$ :
10. Si $f,\varphi$ sont à nème variation bornée il en est de même pour $f+\varphi$ et $cf,c$ étant une constante.

La formule (19) permet de montrer que si $f$ est à nème variation bornée et F à $(n+1)$ ème différence divisée bornée, $\mathrm{F}(f)$ est à nème variation bornée. $f^{k}$ l’est aussi pourvu que $k\geq n+1$ ou bien égal à un nombre entier positif. La propriété est vraie quel que soit $k$ si $|f|>c>0$ . On en déduit que le quotient de deux fonctions à nème variation bornée est à nème variation bornée si le dénominateur reste en module plusgrand qu’un nombre positif.

Il est à remarquer que $f^{k}$ peut ne pas être à nème variatione bornée si $k<n+1$ . Par exemple la fonction

f\left(\frac{1}{n}\right)=\frac{(-1)^{n}}{n^{3}},\quad n=1,2,3,\ldots

est à variation bornée ordinaire (d’ordre 0 ) tandis que $f^{\frac{1}{3}}$ is unbounded in variation.

If the nth total variation of the functions of a family ( $f$ ) remains below a fixed number, any limit function has a total variation of order $n$ at most equal to that number.

Finally, as in No. 7, a sequence of functions $f_{1},f_{2},\ldots$ , $f_{m},\ldots$ , defined respectively on finite sets $\mathrm{E}^{*}{}_{1},\mathrm{E}^{*}{}_{2},\ldots\pi_{\pi}\mathrm{E}^{*}{}_{m},\ldots$ and such

\Delta_{0}\left[f_{m};\mathrm{E}_{m}^{*}\right]\leq\Delta_{0},\quad\mathrm{\penalty 10000\ V}_{n}\left[f_{m};\mathrm{E}_{m}^{*}\right]\leqslant\mathrm{V}_{n}

then there exists at least one limit function defined on the limit set $\mathrm{E}^{*}$ and verifying the inequalities

\Delta_{0}\left[f;\mathrm{E}^{*}\right]\leq\Delta_{0},\quad\mathrm{\penalty 10000\ V}_{n}\left[f;\mathrm{E}^{*}\right]\leq\mathrm{V}_{n}

Formula (22) also shows us, by reasoning analogous to that used in No. 6, that

\Delta_{n}\leq\mathrm{A}\cdot\Delta_{0}+(n+1)\mathrm{V}_{n}

being a fixed number dependent on the sets $\mathrm{E}^{*}{}_{m}$ , this inequality being verified by all functions $f_{m}$ The limit $f(x)$ can then be determined and it also verifies the inequality

\Delta_{n}\left[f;\mathrm{E}^{*}\right]\leq\mathrm{A}.\Delta_{0}+(n+1)\mathrm{V}_{n}

CHAPTER II.

DEFINITION AND MAIN PROPERTIES OF HIGHER-ORDER CONVEX FUNCTIONS.
§. 1. - Classification of functions of a real variable with respect to polynomials.
11. Consider $n+2$ ordered points of the set E

x_{1},x_{2},\ldots,x_{n+2}

(24)

and represent the function $f(x)$ by the points $A_{t}$ coordinates $x_{i},f\left(x_{i}\right),i=1,2,\ldots,n+2$ .

The point $\mathrm{A}_{n+2}$ can have three different positions relative to the representative curve (L) of the polynomial

\mathrm{P}\left(x_{1},x_{2},\ldots,x_{n+1};f\mid x\right)

It can be above, on or below (L). We will say that the function is convex, polynomial, or concave for the points (24) according to the three cases.

Analytically, we will have the three relationships

f\left(x_{n+2}\right)=\mathrm{P}\left(x_{1},x_{2},\ldots,x_{n+1};f\mid x_{n+2}\right)

Formula (2) allows us to write these relations in the form

\left[x_{1},x_{2},\ldots,x_{n,2};f\right]\stackrel{{\scriptstyle>}}{{<}}0

(25)

In this form, we see that the definition is independent of the order of the points.

If the points (24) are ordered, we can also write

\mathrm{U}\left(x_{1},x_{2},\ldots,x_{n+2};f\right)\stackrel{{\scriptstyle>}}{{<}}0

because in this case, $\mathrm{V}\left(x_{1},x_{2},\ldots,x_{n+2}\right)>0$ In general ,
the point $A_{l}$ will have a precise arrangement with respect to the polynomial

\mathrm{P}\left(x_{1},x_{2},\ldots,x_{i-1},x_{i+1},\ldots,x_{n\mid 2};f\mid x\right).

(26)

The function is, for example, convex if the point $A_{i}$ is above or below this line depending on whether $n+2-i$ is even or odd.

We can give the general definition:
The function will be called convex, non-concave, polynomial, non-convex or concave of order $n$ on the set E following the differences divided by order $n+1$ across all groups of $n+2$ points of E $>0,\geq 0,=0,\leq 0,<0$ .

These functions form the class of order functions $n$ .
For $n=0$ , we have monotonic functions. For $n=1$ ordinary convex or concave functions.

If the function $f(x)$ is convex or concave, $-f(x)$ is respectively concave or convex. We can take the non-concave function of order $n$ as a type of order function $n$ Convex and polynomial functions can then be considered special cases. In the study of order functions $n$ Unless otherwise stated, these will always be non-concave functions.

It can happen that a function possesses several convexity properties of different orders. We will say that it belongs to the class ( $a,b,c,\ldots$ ) if it possesses order properties $a,b,c,\ldots$ To highlight the nature of the function, we will assign numbers $a,b,c,\ldots$ clues in the following manner: $a,a^{*},a,a^{\prime},a^{\prime*}$ depending on whether the function is non-concave, convex, polynomial, non-convex or concave of order $a$ It is sometimes useful to distinguish functions of invariable sign. For the sake of uniformity in notation, we will agree to call them functions of order -1, and we will assign this number of indices, as above, depending on whether the function remains $>0,\geq 0,=0,\leq 0,<0$ 12.
If we make the change of variables

x=\alpha x^{\prime}+\beta,\quad y^{\prime}=\gamma y+\delta,\quad f_{1}\left(x^{\prime}\right)=\gamma f\left(\alpha x^{\prime}+\beta\right)+\delta

we obtain easily

\left[x_{4}^{\prime},x_{2}^{\prime},\ldots,x_{n+2}^{\prime};f\right]=\frac{\gamma}{\alpha^{n+1}}\left[x_{1},x_{2},\ldots,x_{n+2};f\right]

Therefore, changing the coordinate axes does not change the order of the function. Changing the units on the axes, or shifting the origin, does not change the convexity of the function. The nature of the function does not change if the orientation of both axes is changed, since the function is of even order, or if the orientation of the x-axis is changed, since the function is of odd order. In all other cases, the non-concave (convex) function changes to a non-convex (concave) function.

Let $a^{\prime},b^{\prime}$ , the endpoints of a subset completely inside E. In the following (24) let us take the point $x_{1}$ in the interval
( $a,a$ , $a$ ) closed on the right and $x_{n+2}$ in the meantime ( $b^{\prime},b$ ) closed on the left. The function is, by definition, contained between the two polynomials L

\mathbb{P}\left(x_{1},x_{2},\ldots,x_{n+1};f\mid x\right),\quad\mathrm{P}\left(x_{2},x_{3},\ldots,x_{n+2};f\mid x\right)

Any order function $n(\geq 0)$ is bounded on any subset completely inside the set on which this function is defined.

If E contains its endpoints, we can take $x_{1}=a,x_{n+2}=b$ and we then see that a function of order $n(\geq 0)$ defined on a set containing its endpoints is bounded.

We can also note that if $f(x)$ If a function belongs to a given class on E, it will belong to the same class on every subset of E, provided, of course, that convexity and polynomiality are considered special cases of non-concavity.
13. Let us now consider functions defined on a finite set.

Let us take a function defined on the ordered sequence (3) and use the notation (4); we then see that

The necessary and sufficient conditions for the tonction to be non-concave (convex, polynomial) of order $n$ of (3) are:

\Delta_{n+1}^{i}\geq 0,(>0,=0),\quad i=1,2,\ldots,m-n-1.

These conditions are, by definition, necessary. Let us show that they are sufficient.

It suffices to show that of the hypothesis

\Delta_{n+1}^{1},\Delta_{n+1}^{2}\geqslant 0,\quad(>0,=0)

One can conclude,

\begin{gathered}{\left[x_{1},x_{2},\ldots,x_{i-1},x_{i+1},\ldots,x_{n+3};f\right]\geq 0,(>0,=0)}\\ i=2,3,\ldots,n+1\end{gathered}

Let's construct L polynomials

	$\displaystyle\mathrm{P}\left(x_{1},x_{2},\ldots,x_{n+1};f\mid=x\right),\quad\mathrm{P}\left(x_{2},x_{3},\ldots,x_{n+2};f\mid x\right)$		(27)
	$\displaystyle\mathrm{P}\left(x_{1},x_{2},\ldots,x_{i-1},x_{i+1},\ldots,x_{n+2};f\mid x\right)$		(28)

and let the point $\mathrm{A}_{i}\left(x_{i},f\left(x_{i}\right)\right)$ We can verify from the figure that if the signs corresponded, not the polynomial (28) would have more in common with at least one of the polynomials (27) $n$ points ⁽¹⁵⁾ which can only happen if the three polynomials (27), (28) coincide. There is then a contradiction, which demonstrates the property. By repeating this process
⁽¹⁵⁾ If two polynomials coincide without intersecting, this point counts at least as two points of intersection.
we can reach all the groups of $n+2$ points of (3). The property also results very simply from formula (10).

The function being non-concave of order $n$ on (3) the continuation
(29)

\Delta_{n+1}^{1},\quad\Delta_{n+1}^{2},\ldots,\Delta_{n+1}^{m-n-1}

does not show any change in sign $\left({}^{16}\right)$ Formula (1) then shows that the sequence

\Delta_{n}^{1},\quad\Delta_{n}^{2},\ldots,\Delta_{n_{4}}^{m+n}

is non-decreasing. If $f$ If the sequence is convex, it is increasing, and if it is polynomial, the sequence has all its terms equal.

A polynomial function of order $n$ is also polynomial of order: greater than $n$ , it can only be convex, polynomials; or concave of order $(n-1)\left({}^{17}\right)$ For a diorder function $n$ or also to ardre- $n-1$ It suffices to add an additional condition, as we have shown: this demonstrates the monotonicity of the sequence (30). This condition is highlighted in the following table.

		nature of order $n$ of the function
		$n^{*}$	$n$	$\bar{n}$	$n!$	$n^{\prime*}$
additional property		$\Delta_{n}^{1}>0$	$\Delta_{n}^{\prime}>0$	$\Delta_{n}^{\prime}>0$	$\Delta_{n}^{m-n}>0$	$\Delta_{ii}^{m-n}>0$
	$\frac{(n-1)^{*}}{(n-1)}$	$\Delta_{l}^{\prime}=0$	$\Delta_{n}^{\prime}=0$	impossibility	$\Delta_{n}^{m-n}=0$	$\Delta_{n}^{m-n}=0$
	$(n-1)^{\prime}$	$\Delta_{n}^{m-n}=0$	$\Delta_{m}^{m-n}=0$	impossibility	$\Delta_{n}^{1}=0$	$\Delta_{t}^{\prime}=0$
	$(n-1)^{\prime*}$	$\Delta_{n}^{m-n}<0$	$\Delta_{n}^{m-n}<0$	$\Delta_{n}^{1}<0$	$\Delta_{n}^{\prime}<0$	$\Delta_{n}^{\prime}<0$

Now, for the function to be of order $-1,1,2,\ldots$ , n we need $m$ conditions at most (including a possible additive constant corresponding to the order -1). For the function to be of class: given, it suffices to equate the divided differences $\Delta_{n+1}^{1},\Delta_{n+1}^{2},\ldots,\Delta_{n+1}^{m-n-1}$ , $\Delta_{n}^{i_{n}},\Delta_{n-1}^{i_{n-1}},\ldots,\Delta_{1}^{i_{1}}$ And $f\left(\mathscr{X}_{i_{0}}\right)$ to suitably chosen numbers, $i_{r}$ being equal to 1 or $m-j$ depending on the nature of the class. We can see immediately that the system is still compatible under the stated restrictions, therefore:

There exist functions of a given class with a lead of m points ${}_{\text{p- }}$ provided that this class meets the following conditions:
$1^{0}$ The order condition $n$ Since it is polynomial, all higher-order conditions are polynomial.
( ¹⁶ ) A sequence $\alpha_{1},\alpha_{2},\ldots$ presents a variation in sign between $\alpha_{m},\alpha_{m+1}$ if $\alpha_{m},\alpha_{m+1}<0$ . If $\alpha_{m+1}=\alpha_{m+2}=\cdots=\alpha_{m+k-1}=0$ there is a variation in sign between $\alpha_{m},\alpha_{m+k}$ when $\alpha_{m}.\alpha_{m+k}.<0$ (17 )
It is, moreover, necessarily to beard $n-1$ 20.
The order condition $n$ being the smallest condition for polymomiality, the order condition $n-1$ is convexity or concavity.

Either $f(x)$ an order function $n$ defined on any set. I say that if it is polynomial on an ordered sequence $x_{1},x_{2},\ldots,x_{n+2}$ it will necessarily be polynomial over the entire part of E contained within the closed interval ( $x_{1},x_{n+2}$ ).

Let $x_{1}{}^{\prime},x_{2}{}^{\prime},\ldots,x_{n+2}^{\prime}n+2$ points of E in $\left(x_{1},x_{n+2}\right)$ not necessarily all distinct from $x_{i}$ It is obviously sufficient to show that

\left[x_{1}^{\prime},x_{2}^{\prime},\ldots,x_{n+2}^{\prime};f\right]=0

However, among the points $x_{i}^{\prime}$ there is at least $un$ which is distinct from $x_{i}$ , For example $x_{4}{}^{\prime}$ , suppose $x_{i}<x_{1}{}^{\prime}<x_{i+1}$ .

11. It suffices to consider the two polynomials $L$
$\mathbb{P}\left(x_{1},x_{2},\ldots,x_{t},x_{1}{}^{0},x_{i+1},\ldots,x_{n};f\mid x\right),\mathrm{P}\left(x_{2},x_{3},\ldots,x_{i},x^{0}{}_{1},x_{i+1},\ldots,x_{n+1};f\mid x\right)$
to see that if $\mathrm{A}_{1}{}^{\prime}\left(x_{1}{}^{\prime},f\left(x_{1}{}^{\prime}\right)\right)$ is not found on

\mathrm{P}\left(x_{1},x_{2},\ldots,x_{n+1};f\mid x\right)

The function cannot be of order n. Analytically, the property is immediate by virtue of formula (10).
14. If the function $f(x)$ is of order $n$ On (3) the sequence (29) does not show any sign variations. We can then deduce that the sequence

\Delta_{n-k+1}^{1},\Delta_{n-k+1}^{2},\ldots,\Delta_{n-k+1}^{m-n+k-1}

(

(\{31)

)

presents $k$ Sign variations at most ( ¹⁸ ).
Suppose that $f(x)$ be defined and of order $n$ on E. We will assume that any function defined on less than $n+2$ points of order $n$ and that its convex nature is the most unfavorable for the properties we have in mind.

We will say that two differences divided by any order

\left[\alpha_{1},\alpha_{2},\ldots,\alpha_{r+1};f\right],\left[\beta_{1},\beta_{2},\ldots,\beta_{r+1};f\right]

are consecutive if we have

\alpha_{1}\leq\alpha_{2}\leq\ldots..<x_{r+1}\leq\beta_{1}<\beta_{2}<\ldots<\beta_{r+1}

or generally
$\alpha_{1}<\alpha_{2}<\ldots<\alpha_{1}=\beta_{1}\leqslant\alpha_{i+1}=\beta_{2}<\ldots<\alpha_{r+1}=\beta_{r-i+2}<\beta_{r-i+3}<\ldots<\beta_{r+1}$
Let's consider the points $\alpha,\beta,\gamma$ dans l’intervalle ( $a,b$ ) tels que $a\leq\alpha\leq\beta\leq\gamma\leq b$ . Soit $\mathrm{E}_{1}$ la partie de E comprise dans l’inter-

⁰⁰footnotetext: (18) Cela sésulte du fait que si

a_{2}-a_{1},a_{3}-a_{2},\ldots,a_{m}-a_{m-1}

, présente .

\pi

variations, la suite,

a_{1},a_{2},\ldots,a_{m}

présente

k+1

variations au plus.

valle $(\alpha,\beta)$ et $E_{2}$ la partie de $E$ comprise dans $(\beta,\gamma)$ . Le point $\beta$ appartient à l’un au moins des ensembles $\mathrm{E}_{1}$ et $\mathrm{E}_{2}$ . Nous dirons alors quo $\mathrm{E}_{1},\mathrm{E}_{2}$ sont deux sous-ensembles consécutifs de $\mathbf{E}$ .

Soit maintenant $c$ un point intérieur à l’intervalle ( $a,b$ ). Je disque la fonetion est d’ordre $n-k$ dans le roisinage gauche et dons le voisinage droit de e.

Pour fixer les idées, démontrons la propriété pour le voisinage gauche. Considérons donc l’intervalle ( $a,c$ ) ouvert ò droite. Il faut démontrer qu’on peut trouver un point $c^{\prime}$ à gauche de $c$ tel que dansl’intervalle ( $c^{\prime},c$ ) ouvert à droite la fonction soit d’ordre $n-k$ . Si $c$ n’appartient pas à E’ ou bien si c est point limite seulement de droite la propriété est évidente puisq’il n’y a alors qu’un nombre fini de points à gauche de c. Supposons donc que c soit point limite de gauche de $E$ et supposons que le point $c^{\prime}$ n’existe pas. On peut alors trouver un point $\alpha$ à gauche de $c$ tel que dans l’intervalle ( $\alpha,c$ ) ouvert à droite il existe deux différences divisées $\Delta_{n-k+1}^{(1)},\Delta_{n-k+1}^{(2)}$ non nulles et de signes contraires. Les résultats du No. 13 nous montrent qu’on peut supposer que $\Delta_{n-k+1}^{(1)},\Delta_{n-k+2}^{(2)}$ soient consécutives. Soit $\alpha^{\prime}$ le point le plus proche de $c$ qui intervient dans ces différences divisées. Dansl’intervalle ( $\alpha^{\prime},c$ ) ouvert à droite on peut trouver deux différences divisées consécutives $\Delta_{n-k+1}^{(3)},\Delta_{n-k+1}^{(4)}$ non nulles et de signes contraires. Soit $\alpha^{\prime\prime}$ le point le plus proche de $c$ qui intervient dans ces différences divisées. On continue le procédé jusqu’à ce qu’on arrive au point $\alpha(k+1)$ . Nous avons ainsi une suite de différences divisées con-sécutives

\Delta_{n-k+1}^{(1)},\Delta_{n-k+1}^{(2)},\ldots,\Delta_{n-k+1}^{(2k+1)},\Delta_{n-k+1}^{(2k+2)}

(32)

qui, par construction, présente au moins $k+1$ variations de signes.-
Considérons tous les points qui interviennent dans les différences. divisées (32) et formons la suite (31) correspondante. Gette suite a, pardéfinition au plus $k$ variations de signes. Or il y a contradiction puisque : (32) en est une suite partielle ( ¹⁹ ). L’existence du point $c^{\prime}$ est donc établie.

On démontre de la même manière la propriété pour le koisinago : droit de $c$ .

La propriété est vraie même pour $k=n+1$ .
Il résulte immédiatement de cette propriété qu’on pout décomposear l’ensenible E en un nombre fini d’ensemble consécutifs
⁽¹⁹⁾ Il est clair qu’une suite présente au moins autant de variations de : signes qu’une quelconque de ses suites partielles.
(33)

\mathrm{E}_{1},\mathrm{E}_{2},\ldots,\mathrm{E}_{m},

tel que sur chacun la fonction soit d’ordre $n-k$ .
Les ensembles $\mathrm{E}_{1}$ et $\mathrm{E}_{m}$ peuvent éventuellement être formés par les seuls points $a$ et $b$ et alors ils n’ont pas de point commun avec $\mathrm{E}_{2}$ resp. $\mathrm{E}_{m-1}$ .

Supposons que la décomposition (33) soit faite de manière qu’on ne puisse pas remplacer ces ensembles par un nombre plus petit d’ensembles vérifiant la même propriété. On peut alors supposer que de deux ensembles $\mathrm{E}_{i},\mathrm{E}_{i+1}$ l’un au moins a au moins $n-k+2$ points.

Montrons qu’on peut former une suite de différences divisées

\Delta_{n-k+1}^{(1)},\Delta_{n-k+1}^{(2)},\ldots,\Delta_{n-k+1}^{(m)}

(34)

toutes différentes de zéro et de signes alternés En effet il existe par hypothèse dans $\mathrm{E}_{1}+\mathrm{E}_{2}$ deux différences divisées non nulles et des signes contraires. On peut les supposer conséculives (No. 13) ; soient $\Delta_{n-l+1}^{(1)},\Delta_{n-k+1}^{(2)}$ . De plus on peut toujours supposer qu’ou bien $\Delta_{n-k+1}^{(1)}$ soit dans $\mathrm{E}_{1}$ , ou bien $\Delta_{n-k+1}^{(2)}$ soit dans $\mathrm{E}_{2}$ . On voit alors qu’on peut trouver une différence divisée $\Delta_{n-k+1}^{(3)}$ consécutive à $\Delta_{n-k+1}^{(2)}$ et située dans $\mathrm{E}_{2}+\mathrm{E}_{3}$ telle que $\Delta_{n-k+1}^{(2)}\cdot\Delta_{n-k+1}^{(3)}<0$ . En effet si celà n’était possible pour aucun choix de $\Delta_{n-k+1}^{(1)},\Delta_{n-k+1}^{(2)}$ la fonction serait d’ordre $n-k$ sur $\mathrm{E}_{2}{}_{7}\mathrm{E}_{3}$ . Et ainsi de suite.

Formons la suite (31) correspondant à tous les points qui interviennent dans (34). Cette suite a au plus $k$ variations ; (34) en est une suite partielle, donc, $m-1\leq k$ , d’où $m\leq k+1$ .

On peut donc énoncer la propriété :
Si la fonction $f(x)$ est d’ordre $n$ sur l’ensemble E , on peut décomposer cet ensemble en $k+1$ ensembles consécutifs au plus sur chacun la fonction étant d’ordre $n-k$ .

Cette décomposition peut en général être effectuée d’une infinité de manières. Dans certains cas, par exemple si la fonction est convexe et si E est un intervalle, la décomposition est unique. On peut le démontrer très facilement.

Si $n=1$ la fonction jouit d’une propriété de convexité ordinaire. Une telle fonction se décompose en au plus deux fonctions monotones et en au plus trois fonctions de signe constant. Les ensembles extrèmes de décomposition peuvent se composer effectivement des points $a$ et $b$ seuls. Soit par exemple la fonction

f(0)=f(1)=1,\quad f(x)=x-1,\quad(0<x<1).

Nous avons les décompositions :
si $k=1$ les ensembles sont $\mathrm{E}_{1}$ (point 0 ), $\mathrm{E}_{2}$ (intervalle $0<x\leq 1$ )
si $k=2$ les ensembles sont $\mathrm{E}_{1}$ (point 0 ), $\mathrm{E}_{2}$ (int. $0<x<1$ ), $\mathrm{E}_{3}$ (point 1 ).
15. Démontrons encore la propriété suivante

Toute fonction d’ordre $n(\geq 0)$ définie sur un ensemble fermé alteint son maximum et son minimum.

Démontrons la propriété relative au maximum. Nous savons déjà que la fonction est bornée, il existe donc un nombre A tel que

		$\displaystyle f(x)<\mathrm{A}-\frac{1}{m}\text{ sur }\mathrm{E}$
		$\displaystyle f(x)>\mathrm{A}-\frac{1}{m}\text{ en au moins un point de }\mathrm{E}$

et ceci pour tout nombre $m$ entier et positif.
Soit une suite monotone, par exemple

x_{1}<x_{2}<x_{3}<\ldots\ldots<x_{m}<\ldots\ldots

(35)

telle que $f\left(x_{m}\right)>\mathrm{A}-\frac{1}{m}$ .
A uńe extraction de suite près on peut supposer que la suite (35) soit telle qu’on ait $x-x_{m}<\frac{1}{m},x$ étant le point limite de la suite.

Il suffit évidemment d’examiner le cas où la suite (35) est infinie. On a alors nécessairement $x>x_{m}$ pour tout $m$ .

De l’inégalité

\mathrm{U}\left(x_{1},x_{2},\ldots,x_{n},x_{m},x;f_{1}\geq 0\quad(m>n)\right.

nous déduisons

f(x)\geq f\left(x_{m}\right)+\frac{\mathrm{B}}{m}

B étant une constante finie, done

\mathrm{A}+\frac{1}{m}>f(x)>\mathrm{A}+\frac{\mathrm{B}-1}{m}

d’où

f(x)=\mathrm{A}.

On démontre de la même manière la propriété relative au minimum.

De la définition résulte qu’une fonction convexe (ou concave) d’ordre $n$ ne peut prendre plus de $n+1$ fois la même valeur. Démontrons la propriété suivante :

Une fonction convexe (ou concave) d’ordre n définie sur un ensem–ble dense dans un intervalle atteint son maximum en $\left\lceil\frac{n+3}{2}\right\rceil\left(\right.$ ou $\left.\left\lceil\frac{n+2}{2}\right\rceil\right)$ et son minimum en $\left[\frac{n+2}{2}\right]\left(\right.$ ou $\left.\left|\frac{n+3}{2}\right|\right)$ points au plus, $|x|$ désignant l’entier égal ou immédiatement inférieur à $\alpha$

Pour fixer lés idées, supposons $n=2m$ et démontions la propriété relative au minimum.

Supposons que contrairement à l’énoncé, le minimum A soit atteint aux points ordonnés

x_{1},x_{2},\ldots,x_{m+k+1}\quad m\geq k>0

(36)

-en tout autre point la fonction étant plus grande que A.
On peut toujours intercaler entre les points (36) les points $x_{1}{}^{\prime},x_{2}{}^{\prime},\ldots,x_{m-k+1}^{\prime}$ de E tels que la suite
(37) $x_{1},x_{2},\ldots,x_{2k},x_{1}^{\prime},x_{2k+1},x_{2}^{\prime},x_{2k+2},\ldots,x_{m+k},x_{m-k+1}^{\prime}\cdot x_{m-k+1}$ soit ordonnée.

Développant le premier membre de la relation
$\left[x_{1},x_{2},\ldots,x_{2k},x_{1}^{\prime},x_{2k+1},x_{2}^{\prime},x_{2k+2},\ldots,x_{m+k},x_{m-k+1}^{\prime},x_{m+k+1};f_{1}>0\right.$ nous avons

\sum_{i=1}^{2m+2}(-1)^{i}\left[f\left(\xi_{i}\right)-A\right]\cdot\frac{V_{i}}{V}>0

(38)

Nous avons par hypothèse

f\left(x_{1}\right)=f\left(x_{2}\right)=\cdots=f\left(x_{m+k+1}\right)=\mathrm{A}

U’inégalité (38) devient donc

\left.-\sum_{i=1}^{m-k+1}\mid f\left(x_{i}^{\prime}\right)-\mathrm{A}\right]\frac{\mathrm{V}_{2k+2i-1}}{\mathrm{\penalty 10000\ V}}>0.

(39)

Nous savons que $f\left(x_{i}^{\prime}\right)>\mathrm{A}$ et que $\mathrm{V}>0,\mathrm{\penalty 10000\ V}_{2k+2i-1}>0,i=1,2,\ldots$ , $m-k+1,\left({}^{20}\right)$ ce qui est en contradiction avec l’inégalité (39). La proeprietí est donc démontrée.

On procède de même dans les autres clas.
(²) En effet si la suite $\alpha_{1},\alpha_{2},\ldots,\alpha_{k}$ est ordonnée on a $V\left(\alpha_{1},\alpha_{2},\ldots,\alpha_{k}\right)>0$ .

Si la fonction est convexe et si le nombre des points cù le maximum est atteint est $\left\lceil\frac{n+3}{2}\right\rceil$ , l’une des extrémités $a,b$ ou bien toutes les deux se trouvent parmi ces points suivant que $n$ est pair ou impair. Si le nombre des points où le minimum est atteint est $\left[\frac{n+2}{2}\right]$ , l’une des extrémités se trouve parmi ces points lorsque $n$ est pair.
16. Si $f$ et $\varphi$ sont deux fonctions de même classe, $f+\varphi$ et $c.f$ où $c$ est une constante positive sont c ncore de même classe. On ne peut en général rien dire sur la classe du produit de deux fonctions. La formule (18) permet d’écrire les propriétés suivantes :

onctions	c l a s s e s
$f$	$(-1,01,2,\ldots,n)!\left(-1,0^{\prime},1,2^{\prime},\ldots,n\left(n^{\prime}\right)\right)$	$(-1,0,1.2,\ldots,n)\left\|\left(-1,0^{\prime},1,2^{\prime},.,n\left(n^{\prime}\right)\right)\right\|$
$\varphi$	$(-1,0,1,2,\ldots,n)$	$\left(-1,0^{\prime},1,2^{\prime},..,n\left(n^{\prime}\right)\right)$
$f.\varphi$	$(-1,0,1,2,\ldots,n)$	$\left(-1,0^{\prime},1^{\prime},2^{\prime},\ldots,n^{\prime}\right)$

De même la fonction de fonction $F(f)$ peut s’étudicr avec la formule (19). On a par exemple les propriétés

fonctions

c l a s s e s

f

(0,1,2,\ldots,n)\left|\left(0,1^{\prime},2,3^{\prime},\ldots,n\left(n^{\prime}\right)\right)\right|\left(0,1^{\prime},2,3^{\prime},\ldots,n\left(n^{\prime}\right)\mid\left(0^{\prime},1,2^{\prime},3,\ldots,n^{\prime}(n)\right)\right.

(0,1,2,\ldots,n)|

$\left(0,1^{\prime},2,3^{\prime},\ldots,n\left(n^{\prime}\right)\right)$	$\left(0^{\prime},1,2^{\prime},3,\ldots,n\left(n^{\prime}\right)\right)$	$0,1,2,\ldots,n$
$\mathrm{\penalty 10000\ F}(f)$	$(0,1,2,\ldots,n)$	$\left(0,1^{\prime},2,3^{\prime}\ldots,n\left(n^{\prime}\right)\right)$
$\left(0^{\prime},1,2^{\prime},3,\ldots,n\left(n^{\prime}\right)\right)$	$\left(0^{\prime},1,2^{\prime},3,\ldots,n^{\prime}(n)\right)$

Il est possible d’établir des résultals plus précis. On peut monts er par exemple que si F est de la classe $(0,1)$ et si $f$ est d’ordre $1,\mathrm{\penalty 10000\ F}(t)$ . est aussi d’ordre 1.

On en déduit par exemple que si $1>0$ est de la classe ( 0,1 ’, $2,3^{\prime}\ldots,n\left(n^{\prime}\right)$ ) la fonction $f^{p},0<p<1$ est de la même classe et $\frac{1}{f}$ est de la classe $\left(0^{\prime},1,2^{\prime}3,\ldots,n\left(n^{\prime}\right)\right.$ .

Si ñous considérons la convexité et la polynomialité comme cas particuliers de la non-concavité on peut énoncer la propriété :

La limite d’une suite convergente de fonctions d’une même classe : est une fonction de la même classe.

Nous avons encore la propriété suivante :
Une fonction continue sur l’ensemble fermé $\mathbf{E}$ et d’une classe donnée sur un sous-ensemble $\mathrm{E}^{*}$ , tel que $\mathrm{E}-\mathrm{E}^{*}=\mathrm{E}^{*\prime}$ , est de la même. classe sur E.

Cette proposition est vraie sans restrictions. Elle est immédiate
pour la non-concavité et la polynomialité et elle résulte pcur la con-vexité de la propriété démontrée à la fin du No. 13.

Soit maintenant une suite de fonctions

f_{1},f_{2},\therefore,f_{m},\ldots

(40)

définies sur les ensembles finis $\mathrm{E}_{4}{}^{*},\mathrm{E}_{2}{}^{*},\ldots,\mathrm{E}_{m}^{*},\ldots$ , compris chacun dans le suivant.

Si $\Delta_{0}\left[f_{m};\mathrm{E}_{m}{}^{*}\right]\leq\Delta_{0},\quad\mathrm{\penalty 10000\ V}_{n}\left[f_{m};\mathrm{E}_{m}{}^{*}\right]\leq\mathrm{V}_{n}$ et si les fonctions (40). sont de la même classe il existe une fonction limite $f$ , définie sur l’ensemble limite $\mathrm{E}^{*}$ , vérifiant les conditions signalées au No. 10 , et quis soit de la même classe.

§ 2. - Relations entre les fonctions d’ordre et de classe donnés et les fonctions étudiées au Chap. I.

17.

Supposons que la fonction $f$ soit définie sur l’ensemble $\mathrm{E}_{1}$ , formé par l’ensemble E et un point isolé $\mathfrak{x}$ . On vérifie facilement que si $f$ est à nème différence divisée bornée sur E , elle est aussi à nème différence divisée bornée sur $\mathrm{E}_{1}$ .

Considérons maintenant une fonction d’ordre $n$ sur E. Soient $a^{\prime},b^{\prime}$ les extrémités d’un sous-ensemble complètement intérieur à E. En vertu de a remarque faite nous pourrons supposer qu’il y ait au. moins $n+1$ points $x_{1}^{\prime},x_{2}^{\prime},\ldots,x_{n+1}^{\prime}$ dans l’intervalle ( $a,a^{\prime}$ ) open to the right and at least $n+1$ points $x^{\prime\prime}{}_{1},x^{\prime\prime}{}_{2},\ldots,x^{\prime\prime}{}_{n+1}$ in the meantime ( $b^{\prime},b$ ) open on the left. Let us finally $x_{1},x_{2},\ldots,x_{n+1},n+1$ points of the considered subset. Because the sequence (30) is monotonic, it follows that the difference divided $\left[\mathfrak{x}_{1},\mathfrak{x}_{2},\ldots,\mathfrak{x}_{n+1};f\right]$ is included between the divided differences $\left[x_{1}{}^{\prime},x_{2}{}^{\prime},\ldots,x^{\prime}{}_{n+1};f\right],\left[x_{1}{}^{\prime\prime},x_{2}{}^{\prime\prime},\ldots,x^{\prime\prime}{}_{n+1}{}^{\prime};f\right]$ ; done

Any function of order n esì with nth divided difference bounded on every completely interior subset E.

It follows that any function of order $n(\geq 1)$ on E is continuous on every subset completely interior to E.

If $\mathrm{E}^{*}$ is a completely interior subset of E on a.

\left|\left[x_{1},x_{2},\ldots,x_{n+1};f\right]\right|\leq\Delta_{n}\left[f;\mathbb{E}^{*}\right]=\text{ nombre fini. }

For any sequence of points (3) of $\mathrm{E}^{\prime}$ we deduce from this.

\left.\sum_{i=1}^{m-1}\left|\Delta_{n}^{i+1}-\Delta_{n}^{i}\right|=\left|\sum_{i=1}^{m-1}\left(\Delta_{n}^{i+1}-\Delta_{n}^{i}\right)\right|=\left|\Delta^{\prime}-\Delta_{n}^{m-n}\right|\leq 2\Delta_{n}\mid f;\mathrm{E}^{*}\right]

done:

Any order function $n$ on E is at nth bounded variation on every subset completely inside E.

In the following section, we will establish a converse of this property.
18. Let us consider the functions defined on a finite set (3). It is easy to see that any function defined on this set can be decomposed into the difference of two functions of the class ( $-1,0,1,2,\ldots,n$ ).

Let us indeed state
(41)

f=\phi-\psi

It is clear that we can take $\phi\left(x_{1}\right)>0$ big enough so that we also have $\psi\left(œ_{1}\right)>0$ Then we can take $\phi\left(w_{2}\right)>0$ big enough so that we also have $\left[x_{1},x_{2};\phi\right]>0,\psi\left(x_{2}\right)>0,\left[x_{1},x_{2};\psi\right]>0$ etc…. etc….

The decompositions (41) are obtained by writing

	$\displaystyle{\left[\mathfrak{x}_{i},\mathfrak{x}_{i+1},\ldots,\mathfrak{x}_{i+k};\phi\right]\geq\frac{1}{2}\left\{1\left[\mathfrak{x},\mathfrak{x}_{i+1},\ldots,\mathfrak{x}_{i+k}\right.\right.}$		(42)
	$\displaystyle\left.+\left[\mathfrak{x}_{i},\mathfrak{x}_{i+1},\ldots,x_{i+k};f\right]\right\}$
	$\displaystyle i=1,\quad k=0,1,2,\ldots,n$
	$\displaystyle k=n+1\quad i=1,2,3,\ldots,m-n-1.$

Consider the system

\begin{array}[]{ll}{\left[x_{1},x_{2},\ldots x_{k+1};\phi\right]=\lambda_{k}}&k=0,1,2,\ldots,n\\ {\left[x_{.}x_{i+1},\ldots,x_{i+n+1};\phi\right]=\mu_{i}}&i=1,2,3,\ldots,m-n-1\end{array}

ade $m$ linear equations in $\phi\left(x_{1}\right),\phi\left(x_{2}\right),\ldots,\phi\left(x_{m}\right)$ Taking into account that the sequence (3) is ordered and applying the fundamental formula (1) we easily find that in general $\phi\left(x_{i}\right)$ is a linear and homogeneous expression of $\lambda_{1},\lambda_{2},\ldots,\lambda_{n};\mu_{1},\mu_{2},\ldots,\mu_{m-n-1}$ -whose coefficients are non-negative.

Be it now $\phi^{*}(x)$ any solution to the system of inequalities (42) and $\phi(x)$ the solution qn'on obtains if we replace all the signs $\geq$ by $=$ The system thus obtained is indeed compatible and the solution is completely determined.

From the previous analysis, we deduce that we have

\phi^{*}\left(\mathfrak{x}_{i}\right)\geqslant\phi\left(\mathfrak{x}_{i}\right)\quad i=1,2,3,\ldots,m.

It is also easy to see that all the equalities can only take place at the same time if (42) is a system of equalities.

The decomposition (41) can be done in an infinite number of ways.

Among these decompositions he $y$ has one for which the functions $\phi,\psi$ . are the smallest possible (*1). From the above, it follows that this canonical decomposition is obtained by the formulas

f(x)=\phi(x)-\psi(x)\quad\text{ sur }(3)

(43) $\left[\mathfrak{x}_{i},\mathfrak{x}_{i+1},\ldots,\mathfrak{x}_{i+k};\phi\right]=\frac{1}{2}\left\{\left|\left[\mathfrak{x}_{i},\mathfrak{x}_{i+1},\ldots,\mathfrak{x}_{i+k};f\right]\right|+\right.$

\begin{gathered}\left.\quad+\left[x_{i},x_{i+1},\ldots,x_{i+k};f\right]\right\}\\ i=1,\quad k=0,1,2,\ldots,n\\ k=n+1,\quad i=1,2,3,\ldots,m-n-1.\end{gathered}

This decomposition has the following properties:
$1^{0}$ . There $(n+1)ee$ function limits $\phi$ And $\psi$ is at most equal to the calle of $f$ 20.
The total variation of order $n$ functions $\phi$ And $\psi$ does not exceed that of $f$ 30.
The functions $\phi,\psi$ are bounded by a quantity dependent on only the properties up to the order $n$ of $f$ and of an interval which contains the points (3), but not of the number of these points.
$1^{0}$ , And $2^{0}$ are immediate. To demonstrate property 30, note that by denoting by $\Delta_{k},\mathrm{\penalty 10000\ V}_{k}$ the limits and total variations: of $f$ we have

		$\displaystyle\left\|\left[x_{1},x_{2},\ldots,x_{k};\phi\right]\right\|\leq\Delta_{k-1}\quad k=2,\ldots,n$
		$\displaystyle\left\|\left[x_{i},x_{i+1},\ldots,x_{i+n};\phi\right]\right\|\leq\frac{1}{2}\left(\mathrm{\penalty 10000\ V}_{n}+2\Delta_{n}\right)\quad i=2,\ldots,m-n$

what we obtain by suitably adding relations (43). Taking (1) into account, we deduce

\begin{gathered}\left|x_{1},x_{2},\ldots,x_{k};\phi_{1}\right|\leq\Delta_{k-1}\quad k=1,2,\ldots,n-1\\ \left.\left|\left[x_{1},x_{i+1},\ldots,x_{i+n-1};\phi\right]\right|<\Delta_{n-1}+\frac{n}{2},b-a\right)\left(\mathrm{V}_{n}+2\mathrm{\penalty 10000\ V}_{n}\right)\\ i=2,3,\ldots,m-n+1\end{gathered}

Or ( $a,b$ ) is an interval containing the points (3). By repeating co: process we arrive at the limitation

|\phi|<\sum_{i=0}^{n-1}i!(b-a)^{i}\Delta_{i}+n!(b-a)^{n}\cdot\frac{1}{2}\left(\mathrm{\penalty 10000\ V}_{n}+2\Delta_{n}\right)\quad\text{ sur (3) }

⁽²¹⁾ A function $f$ is "smaller" than another function $f_{1}$ if we $\mathbf{a}_{-}t\leq f_{1}$ on the common set of the two sets where the functions are defined, the sign < taking place for at least one value of the variable.

This formula also applies to $\psi$ It is rather crude, but sufficient to demonstrate the property $3^{0}$ since it does not deprive Mr.

Let us now consider any function $f$ to nth bounded variation on a set $\mathbf{E}$ and suppose $n>0$ Since the function is continuous, it is completely determined by its values on a countable subset. $\mathrm{E}^{*}$ such as $\mathrm{E}-\mathrm{E}^{*}$ belongs to the derivative of $\mathrm{E}^{*}$ .

Let us now consider E* as the limit of a sequence of finite sets $\mathrm{E}^{*}{}_{1},\mathrm{E}^{*}{}_{2},\ldots\mathrm{E}^{*}{}_{m},\ldots$ each containing the previous one and are

f=\phi_{m}-\psi_{m}\quad m=1,2,\ldots

The minimal decompositions on these sets.

\left\{\begin{array}[]{l}\phi_{1},\phi_{2},\ldots,\phi_{n},\ldots\\ \psi_{1},\psi_{2},\ldots,\psi_{m},\ldots\end{array}\right.

are also bounded, of the class $(-1,0,1,2,\ldots,n)$ and their total variation of order $n$ does not exceed $\mathrm{V}_{n}[f;\mathrm{E}]$ We then know that the functions $\phi_{i}$ have at least one limit $\phi$ on $\mathrm{E}^{*}$ and consequently the $\psi_{i}$ also have a limit $\psi$ such as

\begin{gathered}\mathrm{V}_{n}\left[\phi;\mathrm{E}^{*}\right]\leq\mathrm{V}_{n}[f;\mathrm{E}],\quad\mathrm{V}_{n}\left[\psi;\mathrm{E}^{*}\right]\leq\mathrm{V}_{n}[f;\mathrm{E}]\\ \phi,\psi\text{ de la classe }(-1,0,1,2,\ldots,n)\\ f=\phi-\psi\quad\text{ sur }\mathrm{E}^{*}\end{gathered}

By the principle of continuity, we deduce the following property:
Every function with bounded nth variation is the difference of two functions of the class $(1,0,1,2,\ldots,n)$ and whose nth total variation does not exceed that of $\left.f(\tau){}^{(22}\right)$ .

Either $f=\phi-\psi$ the minimum decomposition on the sequence (3). Let's add a new point to points (3). $x_{0}$ and either $f_{1}=\phi_{1}-\psi_{1}$ the minimum decomposition on the new sequence obtained ( $f_{1}=f$ on (3)). It is easily shown using formulas (43) that we have $\phi\leq\phi_{1}$ on (3) (23). We deduce that the sequences (44) do indeed have a limit.

It follows that the decomposition obtained above for any set also satisfies the minimum property, that is to say-

⁰⁰footnotetext: (22) Pour le cas

n=1

M. A. Winternitz (voir loc. cit. 5) a démontré que toute fonction à première variation bornée (Funktion von beschränkter Drehung) est la différence de deux fonctions d’ordre 1. La méthode de cet -auteur est différente de celle employée ici.
(23) L’égalité peut d’ailleurs avoir lieu partout. Si

x_{i}<x_{0}<x_{i+1}

on a certaimement

\phi\left(x_{j}\right)=\phi_{1}\left(x_{j}\right),j=1,2,\ldots,i

to say that among all the functions $\phi,\psi$ class $(-1,0,1,2,\ldots,n)$ verifying equality $f=\phi-\psi$ Those obtained above are the smallest possible. Otherwise, in fact, it would show that there exists a $m_{m}$ for which $f=\phi_{m}-\psi_{m}$ is not the minimum decomposition on $\mathrm{E}^{*}{}_{m}$ -which is impossible.

We assumed $n>0$ . If $n=0$ the function $f$ is not generally continuous, but its points of discontinuity form a countable set. It suffices to take the set E* such that it contains the set of discontinuities; the method is then applicable and necessarily leads to the minimum decomposition into two non-negative and non-decreasing functions, which is well known.

CHAPTER III.

ON THE DERIVATIVES OF THE FUNCTIONS OF A REAL VARIABLE.

§ 1. - Some remarks on the definition
of the derivative of order $n$ .

19.

We will now assume that the function $f(x)$ either decorated on E. When the points $x_{1},x_{2},\ldots,x_{n_{T1}}$ tend towards a point $x$ of $\mathrm{E}^{\prime}$ the difference divided

\left[x_{1},x_{2},\ldots,x_{n+1};f\right]

(45)

does not generally tend towards any limit.
We will designate by $f_{n}(x)$ the limit of the divided difference (45), if it exists, is finite and well-determined when the points $x_{1},x_{2},\ldots,x_{n+1}$ tend in some way towards $x$ .

Suppose that the point $x$ belongs to E, we will then designate by $f_{n}{}^{*}(x)$ the limit of the difference divided $\left[x,x_{1},x_{2},\ldots,x_{n};f\right]$ if it exists, is finite and well-defined when the points $x_{1},x_{2},\ldots,x_{n}$ tend in some way towards $x$ .

It can easily be demonstrated that in order that $f_{n}(x)$ exists, it is necessary and sufficient that for every number $\varepsilon>0$ we can match a number $\eta>0$ such as
(46) $\quad\left[x_{1},x_{2},\ldots,x_{n+1};f\right]-\left[x_{1}^{\prime},x_{2}^{\prime},\ldots,x_{n+1}^{\prime};f\right]\mid<\varepsilon$ , In $\mathrm{I}(x;\eta){}^{(24)}$ .

This inequality can also be replaced by the following (formula (6)):
(47) $\quad\left|\left[x_{1},x_{2},\ldots,x_{n+1};f\right]-\left[x_{2},x_{3},\ldots,x_{n+2};f\right]\right|<\varepsilon$ , In $\mathrm{I}(x;\eta)$
( ²⁴ ) $\mathrm{I}(x;\eta)$ denotes the midpoint interval $x$ and length $\eta$ .

We can also take $\eta>0$ so that

\left|\left[x_{1},x_{2},\ldots,x_{n+1};f\right]-f_{n}(x)\right|<\varepsilon,\quad\text{ dans }\mathrm{I}(x;\eta)

(48)

The same properties hold true for $f_{n}^{*}(x)$ by including* the point $x$ in all the divided differences of formulas (46), (47) and (48).

Formula (9) shows that if $f_{n}^{*}(x)$ exists, the divided difference (45) tends towards $f_{n}{}^{*}(x)$ when the points $x_{1},x_{2},\ldots,x_{n+1}$ tend towards $x$ provided that the quotient

\left|\frac{x-x_{1}}{x_{1}-x_{2}}\right|

remains bounded. Indeed, in this case, any limit of (45) is finite, and from any sequence of divided differences converging to a limit, we can extract a new sequence such that

\frac{x-x_{1}}{x_{1}-x_{2}}

has a limit, which is finite by hypothesis. Formula (9) then shows that this sequence of divided differences tends towards $f_{n}{}^{*}(x)\left({}^{25}\right)$ .

We will assume that E is closed.
The existence of $f_{n}(x)$ entails by definition that of $f_{n}{}^{*}(x)$ By definition ,
the $n$ th derivative $f^{(n)}(x)$ is the derivative of the $(n-1$ the th derivative and, by definition, also the first derivative $f^{\prime}(x)$ is identical to $f_{1}{}^{*}(x)$ .
$f^{(n)}(x)$ can therefore be defined on the set $\mathrm{E}^{(n)}$ and it is evident that its existence implies that of $f^{(n-1)}(x)$ in all aspects of $\mathrm{E}^{(n-1)}$ in a neighborhood $\mathrm{I}(x;\eta),\eta>0$ of $x$ When we talk about the continuity of $f^{(n)}(x)$ at one point $x$ of $E^{(n)}$ we assume, of course, that this derivative exists in the neighborhood of $x$ 20.
From (46) we deduce that if $f_{n}(x)$ exists at a point $x$ of : $\mathrm{E}^{\prime}$ the function is a divided nth difference function bounded in the neighborhood of $x$ If the function is at $(n+1)$ nth difference divided bounded, $f_{n}(x)$ exists at every point of $\mathrm{E}^{\prime}$ and is continuous (on $\mathrm{E}^{\prime}$ ). If $f_{n}(x)$ exists in. every point of $\mathrm{E}^{\prime}$ It is continuous, therefore necessarily bounded. It follows that the function has a bounded nth difference. In this case, inequality (46) holds uniformly on E.

If $f_{n}(x)$ exists at a point $x,f_{n-1}(x)$ also exists at this point and:
(25) This remark allows us to define $f_{n}{}^{*}(x)$ even at a point of $\mathrm{E}^{\prime}$ which does not belong to E, but it is unnecessary to consider this extension.
in its neighborhood, but it should be noted that $f_{n}(x)$ can exist at a point (of $E^{\prime\prime}$ ) without existing in its vicinity ( ²⁶ ).

If $f_{n}^{*}(x)$ exists at a point $x,f_{n-1}^{*}(x)$ also exists at this point, but not generally in its vicinity ( ²⁷ ).

Finally $f_{n}^{*}(x)$ can exist at a point and even be continuous without $f_{n}(x)$ exists ( ²⁸ ).

We can demonstrate the following property:
(26) Let the function

\begin{array}[]{ll}f(0)=0,f\left(\frac{1}{n}\right)=0,&n=2,3,\ldots\\ f\left(\frac{1}{n+1}+\frac{1}{3^{p}n(n+1)}\right)=0,&p=1,2,\ldots n=2,3,\ldots\\ f\left(\frac{1}{n+1}+\frac{1}{2\cdot 3^{p}n(n+1)}\right)=\frac{1}{2\cdot 3^{p-1}n^{2}(n+1)},&p=1,2,\ldots n=2,3,\ldots\end{array}

The whole $E^{\prime}$ is formed by the points

\frac{1}{2},\frac{1}{3},\ldots,\frac{1}{n},\ldots,\text{ et }0

At the points $\frac{1}{2},\frac{1}{3},\ldots,f_{1}(\mathcal{x})$ does not exist, on the contrary $\left|\left[x_{1},x_{2};f\right]\right|\leq\frac{1}{n}$ In $\left(0,\frac{1}{n}\right)$ SO $f_{1}(0)$ exists and is equal to zero.
(27) Consider the function defined for $0\leq x\leq\frac{1}{2}$ in the following way
$f(0)=0,.f(x)=\frac{1}{2^{n+1}n}\left(x-\frac{1}{2^{n+1}}\right)$ For $\frac{1}{2^{n+1}}\leq x\leq\frac{3}{2^{n+2}},n=1,2,3,\ldots$

f(x)=-\frac{1}{2^{n+1}n}\left(x-\frac{1}{2^{n}}\right)\quad\text{ pour }\frac{3}{2^{n+2}}\leq x\leq\frac{1}{2^{n}}\quad n=1,2,3,\ldots

We check that $f_{2}^{*}(0)$ exists and is equal to zero, while $f_{2}^{*}(x)$ obviously does not exist at points $\frac{1}{2^{2}},\frac{1}{2^{3}},\frac{1}{2^{4}},\ldots$
⁽²⁸⁾ It suffices to take the function

\begin{array}[]{lr}f(0)=0,f\left(\frac{1}{n}\right)=0,&n=2,3,\ldots,\\ f\left(\frac{1}{n+1}+\frac{1}{2np(n+1)}\right)=0,&p=1,2,\ldots,n=1,2,\ldots,\\ f\left(\frac{1}{n+1}+\frac{1}{(2p+1)n(n+1)}\right)=\frac{1}{p^{2}n^{\prime}(n+1)},p=1,2,\ldots,n=1,2,\ldots,\end{array}

$f_{1}^{*}(x)$ exists in every respect $0,\frac{1}{2},\frac{1}{3},\ldots$ , but $f_{1}(0)$ does not exist.

If $f_{n}(x)$ exists at a point of $\mathrm{E}^{(n)}$ the nth derivative $f^{(n)}(x)$ exists at this point and we have equality:

f^{(n)}(x)=n!f_{n}(x)

Indeed, there is a number $\eta>0$ such as in $\mathrm{I}\left(x;\gamma_{1}\right)$ the nth difference divided by $f(x)$ does not exceed the finite number A in modulus. All limits $f_{1}(x),f_{2}(x),\ldots,f_{n-1}(x)$ exist in $\mathrm{I}(x;\eta)$ We can then prove the property by induction. It is obvious for $n=1$ Let us therefore assume that it is true for $1,2,\ldots,n-1$ and let's take it apart to $n$ To all $\varepsilon>0$ corresponds to a $\eta_{1}\leq\eta$ such that if $x^{\prime}$ is a point of $\mathrm{E}^{(n-1)}$ In $\mathrm{I}\left(x;\eta_{1}\right)$ and if $x_{1}{}^{\prime},x_{2}{}^{\prime},\ldots,x_{n}{}^{\prime}$ are close enough to $x^{\prime}$ And $x_{1},x_{2},\ldots,x_{n}$ sufficiently close to $x$ we have at the same time

\begin{gathered}\left|\left[x_{1}^{\prime},x_{2}^{\prime},\ldots,x_{i}^{\prime},x_{i},x_{i+1},\ldots,x_{n};f\right]-f_{n}(x)\right|<\frac{\varepsilon}{3n},\left|1-\frac{x_{i}^{\prime}-x_{i}}{x^{\prime}-x}\right|<\frac{\varepsilon}{3n\mathrm{\penalty 10000\ A}}\\ i=1,2,\ldots,n\\ \left|\left[x_{1},x_{2},\ldots,x_{n}\right]-f_{n-1}(x)\right|<\frac{\left|x^{\prime}-x\right|}{6}\varepsilon\\ \left|\left[x_{1}^{\prime},x_{2}^{\prime},\ldots,x_{n}^{\prime};f\right]-f_{n-1}\left(x^{\prime}\right)\right|<\frac{\left|x^{\prime}-x\right|}{6}\varepsilon\end{gathered}

But, hypothetically

f_{n-1}(x)=\frac{1}{(n-1)!}f^{(n-1)}(x),\quad f_{n-1}(x)=\frac{1}{(n-1)!}f^{(n-1)}\left(x^{\prime}\right)

We have done
$\left|\frac{1}{(n-1)!}\frac{f^{(n-1)}(x)-f^{(n-1)}\left(x^{\prime}\right)}{x-x^{\prime}}-\frac{\left[x_{1},x_{2},\ldots,x_{n};f\right]-\left[x_{1}{}^{\prime},x_{2}{}^{\prime},\ldots,x_{n}{}^{\prime};f\right]}{x-x^{\prime}}\right|<\frac{\varepsilon}{3}$

\left|nf_{n}(x)-\sum_{i=1}^{n}\frac{x_{i}-x_{i}^{\prime}}{x-x_{i}}\left[x_{1}^{\prime},x_{2}^{\prime},\ldots,x_{i}^{\prime},x_{i},\ldots,x_{n};f\right]\right|<\frac{2\varepsilon}{3}

Formula (6) gives

\left|n!f_{n}(x)-\frac{f^{(n-1)}(x)-f^{(n-1)}\left(x^{\prime}\right)}{x-x^{\prime}}\right|<\varepsilon\quad\text{ dans }\mathrm{I}\left(x;\eta_{1}\right)

which demonstrates the property.
We deduce that so that $f^{(n)}(x)$ exists and either continues on $E^{(n)}$ it is enough that $f_{n}(x)$ exists on $E^{(n)}$ , but it should be noted that this condition is not generally necessary.

Let us also note that, from the existence of $f_{n}^{*}(x)$ , we cannot generally conclude that of $f^{(n)}(x)(n>1)$ 21.
More precise properties can be obtained if E is a closed win interval ( $a,b$ ). Stieltjes demonstrated ( ²⁹ ) in fact that:

If $f^{(n)}(x)$ exists and is continuous at the point $x,f_{n}(x)$ exists and we have

f_{n}(x)=\frac{f^{(n)}(x)}{n!}

Comparing this with what was said above, we see that:
The necessary and sufficient condition for a function $f(x)$ ubéfinite and bounded in the interval ( $a,b$ ) has a continuous nth derivative - in this interval is that the divided nth difference is uniformly continuous in ( $a,b$ ).

We already know that the nth difference divided is uniformly continuous if at every number $=\varepsilon>0$ and in every respect $x$ of $(a,b)$ we can match a number $\eta>0$ such that one has
$\because\mid\left[x_{1},x_{2},\ldots,x_{n+1};f\left|-\left[x_{2},x_{3},\ldots,x_{n+2};f\right]\right|<\varepsilon\operatorname{dans}\mathrm{I}(x;\eta)\right.$
In the case of an interval, if $f_{n}{}^{*}(x)$ exists and is continuous at the point $x,f_{n}(x)$ also exists at this point. Let us suppose, in fact, that in $\mathbb{I}(x;\delta),\delta>0,f_{n}^{*}(x)$ exists and either continues to the point $x$ To all $\varepsilon>0$ corresponds to a $\eta,(0<\eta\leq\delta)$ such as

\left|f_{n}^{*}(x)-f_{n}^{*}\left(x^{\prime}\right)\right|<\varepsilon\quad\text{ dans }\mathrm{I}(x;\eta)

f_{n}^{*}\left(x_{n}^{\prime}\right)=\left[x_{1},x_{2},\ldots,x_{n+1};f\right]

\left|f_{n}^{*}(x)-\left[x_{1},x_{2},\ldots,x_{n+1};f\right]\right|<\varepsilon\quad\text{ dans }I(x;\eta)

$f_{n}(x)$ exists and is obviously equal to $f_{n}{}^{*}(x)$ It is important
to note that even in the case of an interval $f_{n}(x)$ (Or $f_{n}*(x)$ ) can exist at a point without existing in its neighbor (image (30)). Similarly if $f_{n}{}^{*}(x)$ exists to the point $x,f_{n-1}^{*}(x)$ also exists at the point $x$ , but not generally in its neighborhood ( ³¹ ).
(29) TJ Stieltjes „Over Lagrange's interpolatie-formulae" Verslagen cen Mendeelingen der Koninklijke Akademie van Wetenschappen te Ansterdam 2nd ser. t. XVII (1882), p. 239.
(30) See the example given in note ( ²⁷ ).
(31) The example in note ( ²⁷ ) verifies the property.

Stieltues also proved the following proposition ( ³² ): -
If $f(x)$ is defined and bounded in ( $a,b$ ) and if $f^{(n)}(x)$ exists at a single point, $f_{n}^{*}(x)$ also exists at this point and we have

f_{n}^{*}(x)=\frac{1}{n!}f^{(n)}(x)

This is a necessary condition for the existence of the nth derivative, but it is not sufficient in general. Instead, we will demonstrate the following property:

The necessary and sufficient condition for the function $f(x)$ defined and bounded within the interval ( $a,b$ ) has an nth derivative at every point of ( $a,b$ ) is that $f_{n}{}^{*}(x)$ exists at every point of ( $a,b$ ).

We know that the condition is necessary. Let's show that it is sufficient.

In this case $f^{*}{}_{n-1}(x)$ exists everywhere and is continuous, therefore.

f_{n-1}^{*}(x)=f_{n-1}(x)=\frac{1}{(n-1)!}f^{(n-1)}(x)

It will suffice to show that $f_{n-1}(x)$ has a derivative at every point of ( $a,b$ ). Either $x$ a point of ( $a,b$ ) ; we can find $\mathrm{A}>0$ And $\delta>0$ such as

\left|\left[x,x_{1},x_{2},\ldots,x_{n};f\right]\right|<\mathrm{A}\quad\text{ dans }\mathrm{I}(x;\delta)

$\varepsilon>0$ given, one can find $0<\eta\leq\delta$ , such as
(49) $\left|\left[x,x_{1},x_{2},\ldots,x_{n};f\right]-\left[x,x_{1}^{\prime},x_{2}^{\prime},\ldots,x_{n}^{\prime};f\right]\right|<\varepsilon$ In $I(x;\eta)$ .

Let $x^{\prime}$ a point of $I^{\prime}(x;\eta),x_{2},x_{3},\ldots,x_{n}$ points in the vicinity of $x$ And $x_{1}{}^{\prime},x_{2}{}^{\prime},\ldots,x_{n}{}^{\prime}$ points in the vicinity of $x^{\prime}$ We can find the points $x^{*},x^{**}$ In $\mathrm{I}(x;\eta)$ such $q\mu e$
$f_{n-1}\left(x^{*}\right)=\left[x,x_{2},x_{3},\ldots,x_{n};f\right],f_{n-1}\left(x^{*}\right)=\left[x_{1}{}^{\prime},x_{2}{}^{\prime},\ldots x_{n}{}^{\prime};f\right]$ .
Gold, $f_{n-1}(x)$ Since it's continuous, we can choose the points $x_{i},x_{i}^{\prime}$ such _that one has both

\left|\frac{x_{i}-x_{i}^{\prime}}{x^{*}-x^{\prime*}}\right|<1+\varepsilon\quad i=1,2,\ldots,n,\left(x_{1}=x\right)

\left|\frac{f_{n-1}\left(x^{*}\right)-f_{n-1}\left(x^{*}\right)}{x^{*}-x^{\prime*}}-\frac{f_{n-1}(x)-f_{n-1}\left(x^{\prime}\right)}{x-x^{\prime}}\right|<\frac{\varepsilon}{3}.

(50)

Let's consider another point $x^{\prime\prime}$ In $\mathrm{I}(x;\eta)$ , and let's correspond to it*-
⁽³²⁾ TJ STIELTJES „Einige bemerkingen omtrent de differentialquotienten. van eene functie van eene veranderlijke" Niéuw. Archief for Wiskunde t. IX, (I882), p. 106-111.
place the points in the same way $x_{1}{}^{\prime\prime},x_{2}{}^{\prime\prime},\ldots,x_{n}{}^{\prime\prime};x^{\prime\prime*}$ such that the corresponding inequalities (49), (50) are satisfied. It is always possible to take these points in such a way that we also have

\left|\frac{x_{i}-x_{i}^{\prime}}{x^{*}-x^{**}}-\frac{x_{i}-x^{\prime\prime}}{x^{*}-x^{\prime*}}\right|<\frac{\varepsilon}{6n\mathrm{\penalty 10000\ A}},\quad i=1,2,\ldots,n\left(x_{1}=x\right)

Formula (6) then gives us

\begin{gathered}\left\lvert\,\frac{\left[x,x_{2},x_{3},\ldots,x_{n};f\right]-\left[x_{1}^{\prime},x_{2}^{\prime},\ldots,x_{n}^{\prime};f\right]}{x^{*}-x^{**}}-\right.\\ \left.-\frac{\left[x,x_{2},x_{3},\ldots,x_{n};f\right]-\left[x_{1}^{\prime\prime},x_{2}^{\prime\prime},\ldots,x_{n}^{\prime\prime};f\right]}{x^{*}-x^{\prime\prime*}}\right\rvert\,\leq\\ \leq\sum_{i=1}^{n}\left|\frac{x_{i}-x_{i}^{\prime}}{x^{*}-x^{\prime*}}-\frac{x_{1}-x_{i}^{\prime\prime}}{x^{*}-x^{\prime\prime*}}\right|\left|\left[x_{1},x_{2},\ldots,x_{i},x_{i}^{\prime\prime},x_{i+1}^{\prime\prime},\ldots,x_{n}^{\prime\prime};f\right]\right|+\\ \left.+\left|\frac{x_{i}-x_{i}^{\prime}}{x^{*}-x^{\prime*}}\right|\right\rvert\,\left[x,x_{2},x_{3},\ldots,x_{i},x_{t}^{\prime},\ldots,x_{n}^{\prime};f\right]-\\ \left.-\left[x,x_{2},x_{3},\ldots,x_{i},x_{i}^{\prime\prime},\ldots,x_{n}^{\prime\prime};f\right]\mid\right\}<\frac{\varepsilon}{3}\end{gathered}

\left|\frac{f_{n-1}(x)-f_{n,1}\left(x^{\prime}\right)}{x-x^{\prime}}-\frac{f_{n-1}(x)-f_{n-1}\left(x^{\prime\prime}\right)}{x-x^{\prime\prime}}\right|<\varepsilon\quad\text{ dans }\mathrm{I}(x;\eta)

"which demonstrates the property.
22. Now suppose that $f(x)$ either bounded and summable in ( $a,b$ The function

\mathrm{F}_{a}(x)=1_{a}^{(\alpha)}f(x)=\frac{1}{\Gamma(\alpha)}\int_{a}^{x}(x-t)^{\alpha-1}f(t)dt

is then continuous for $\alpha>0$ We will demonstrate the more general property (always assuming $\alpha>0$ ) :

If $f(x)$ is a bounded divided nth difference, the divided nth difference of the function $\mathrm{F}_{a}(x)$ is uniformly continuous in every interval $\left\{\left(a_{1},b\right),a_{1}>a_{0}\right.$ .

We know that this property means that at any $\varepsilon>0$ and in every respect $x$ of $\left(a_{1},b\right)$ we can match a $\eta>0$ such that we have (51) $l_{1}\left[x_{1},x_{2},\ldots,x_{n+1};\mathrm{F}_{a}\right]-\left[x_{2},x_{3},\ldots,x_{n+2};\mathrm{F}_{a}\right]\mid<$ In $\mathrm{I}(x;\eta)$
Let's make the change of variables .

t\mid(x-a)t+a

which is legitimate ( ³³ ) and let's remove the factor $\frac{1}{\Gamma(\alpha)}$ We can consider, with a slight change in notation,

\mathrm{F}_{a}^{*}(x)=\int_{0}^{1}(1-t)^{a-1}(x-a)^{a}f((x-a)t+a)dt

We have (formula (8)):

\left[x_{1},x_{2},.,x_{n+2};(x-a)^{a}f((x-a)x+a)\right]=

(52)

\sum_{k=1}^{n+2}\left[x_{1},x_{2},\ldots,x_{k};(x-a)^{a}\right].\left[x_{k},x_{k+1},\ldots,x_{k+2};f((x-a)t+a)\right]

However, the function $(x-a)^{a}$ and all its divided differences are bounded in the interval ( $a_{1},b$ ). The same applies, by hypothesis, to the function: $f((x-a)t-+a)$ until order $n$ included in ( $a_{1},b$ ) and for all values of $t$ It therefore suffices to demonstrate that for any number $\varepsilon>0$ we can match a number $\eta>0$ such that one has
$\left|\int_{0}^{1}\left(1-t_{j}{}^{a-1}\left\{\left[x_{1},x_{2},\ldots,x_{n+1};f((x-a)t+a)\right]-\left[x_{2},x_{3},\ldots,x_{n+2};f(x-a)t+a\right)\right]\right\}dt\right|<\varepsilon$ provided that $x_{1},x_{2},\ldots x_{n+2}$ remain within a length interval $\eta$ .

The general case follows immediately from the case $n=0$ We know, in fact, that $f^{(n-1)}(x)$ exists and is continuous. Furthermore, we can find a point $\xi$ In $(a,b)$ and included in the smallest interval containing the points $x_{1},x_{2},\ldots,x_{n+1}$ such that in any interval containing the point $\xi$ there exists a difference divided equal to $\left[x_{1},x_{2},..,x_{n+1};f((x-a)t+a)\right]$ This point $\xi$ se trouve toujours dans le plus petit intervalle où les points, sur lesquels ces différences divisées sont prises, sont situés. Supposonsque $f(x)$ ait une nème dérivée au point $(\xi-a)t+a$ , alors en vertu du second théorème de Stifltjes

\left[x_{1},x_{2},\ldots,x_{n+1};f((x-a)t+a)\right]=\frac{t^{n}f^{(n)}((\xi-a)t+a)}{n!}

De même nous pouvons trouver $\xi^{\prime}$ de manière que si $f(x)$ a une : nème dérivée au point $\left(\xi^{\prime}-a\right)t+a$ , on a

\left[x_{2},x_{3},\ldots,x_{n+2};f((x-a)t+a)\right]=\frac{t^{n}f^{(n)}((\xi-a)t+a)}{n!}

⁰⁰footnotetext: ⁽³³⁾ Voir H. Lebesgue, "Sur les intégrales singulières", Ann. Fac. Toulouse 3ème s. t. L. (1909), pp. 25-117, sp. p. 44.

Les points $\xi,\xi^{\prime}$ sont dans un intervalle de longueur $\leq\eta$ . D’autre part $f^{(n-1)}(x)$ existe partout et est à première différence divisée bornée, en vertu d’un théorème de M. Lebesgue $f^{(n)}(x)$ existe donc presque partout et est évidemment bornée. En vertu des propriétés bien connues des fonctions sommables il résulte que le prob’ème se réduit à la continuité d’une expression de la forme $F_{a}(x)$ . Ce que nous savonsêtre exact ( ³⁴ ).

On sait que la dérivée d’ordre $\alpha(\alpha>0)$ est définie par l’égalité :

\left.\mathrm{D}^{(a)}f(x)=\frac{d^{n}}{dx^{n}}\mathrm{I}_{a}^{\prime n-a)}f(x)=\frac{d^{n}}{dx^{n}}\mathrm{\penalty 10000\ F}_{n-a^{\prime}}x\right)

$n$ étant entier $>\alpha$ .
La première proposition énoncée au No. 21 nous permet d’éerire les conditions nécessaires et suffisantes pour l’existence d’une dérivée continue d’ordre $\alpha$ . En vertu de la propriété exprimée par la formule (51) nous déduisons que si $t(x)$ est à nène différence divisée bornée la. dérivée d’ordre $\alpha$ existe et est continue dans $(a,b)$ pour $\alpha<n$ . C’est un théorème de M. P. Montel, énoncé d’une façon un peut différente. M. Montel a moniré que, la fonction étant supposée bornée, il suffit de considérer seulement les différences divisées pises sur des points équidistants ( ³⁵ ) :

Les propriétés démontrées au No. 21 permettent d’écrire lés conditions nécessaires et suffisantes pour l’existence, dans l’intervalle ( $a,b$ ) ouvert à gauche, de la dérivée d’ordre $\alpha$ ou bien de la dérivée continue d’ordre $\alpha$ . Pratiquement ces enoncés ne présentent pas beaucoup d’intérêt. Il est possible par diverses transformations d’en déduire les critères donnés par M. Marchaud. Nous n’msistons pas sur cette question qui nous eloignerait trop de notre sujet et nous renverrons au memorie de M. A. Marchaud ( ³⁶ ).

§. 2. - Remarques sur les dérivées des fonctions étudiées aux Chap. I. et II.

23.

Nous déduisons de ce qui précède qu’une fonction d’ordre n sur E a des dérivées continues d’ordre $1,2,\ldots,n-1$ sur tout sousensemble complèlement intérieur à E. Si la fonction est définie dans un intervalle elle a des dérivées continues d’ordre $r<n$ dans tout intervalle complètement intérieur.
(34) A la rigueur ceci ne résulte que si $x$ est dans $\left(a_{1},b\right)$ ce qui n’est pas en contradiction avec notre hypolhèse initiale.
(35) P. Montel "Sur les polynomes d’approximation" Bull. de la Soc. Math. t. 46 (1918) pp. 151-192 sp. p. 183.
(36) Voir loc, cit. (7).

Posons

\mathrm{U}\left(\begin{array}[]{l}x_{1},x_{2},\ldots,x_{n+1}\\ x_{1}^{\prime},x_{2}^{\prime},\ldots,x_{n+1}^{\prime}\end{array};f\right)=

$=\left|\begin{array}[]{cccc}\left.1\mid x_{1},x_{1}{}^{\prime};x^{2}\right]&{\left[x_{1},x_{1}{}^{\prime};x^{3}\right]}&\ldots\left[x_{1},x_{1}{}^{\prime};x^{n}\right]&{\left[x_{1},x_{1}{}^{\prime};f\right]}\\ 1\left[x_{2},x_{2}{}^{\prime};x^{2}\right]&{\left[x_{2},x_{2}{}^{\prime};x^{3}\right]}&\ldots\left[x_{2},x_{2}{}^{\prime};x^{n}\right]&{\left[x_{2},x_{2}{}^{\prime};f\right]}\\ \cdots\cdots\cdots\cdots\cdots\cdots&\cdots\cdots&\cdots\\ 1\left[x_{n+1},x_{n+1}^{\prime};x^{2}\right]&{\left[x_{n+1},x_{n+1}^{\prime};x^{3}\right]\ldots\left[x_{n+1},x_{n+1}^{\prime};x^{n}\right]}&{\left[x_{+1},x_{n+1}^{\prime};f\right]}\end{array}\right|$
Cette expression s’annule identiquement lorsque $f(x)$ est un polynome de degré n. Done, pourvu que tous les points

x_{1},x_{1}^{\prime},x_{2},x_{2}^{\prime},\ldots,x_{n+1},x_{n+1}^{\prime}

(53)

soient distincts, elle est nécessairement de la forme
(54) $\left.\mathrm{U}\left(\begin{array}[]{l}x_{1},x_{2},\ldots,x_{n+1}\\ x_{1}^{\prime},x_{2}^{\prime},\ldots,x_{n+1}^{\prime}\end{array};f\right)=\sum_{i=1}^{n}\mathrm{\penalty 10000\ A}_{1}.\mid y_{i},y_{i+1},\ldots,y_{i+n+1};f\right]$

y_{2i+1}=x_{i},y_{2i}=x_{i}^{\prime},\quad i=1,2,\ldots,n+1

This formula can be determined as follows: Subtract each row of the determinant from the next. Then, applying formula (6), the determinant decomposes (by virtue of Binet's formula giving the product of two tables) into a sum of determinants of order $n$ of the same form but relating to divided second differences, each multiplied by a factor independent of the function $f$ We decompose each determinant in the same way until we arrive at formula (54). If we assume that the sequence (53) is ordered, we can write formula (6) so that the factors introduced are always positive, and this process effectively leads to (54). It follows that if the sequence (53) is ordered, the coefficients $\mathrm{A}_{i}$ in (54) are positive.

If the derivative $f^{\prime}(x)$ exists and is continuous, we have

\lim\mathrm{U}\left(\begin{array}[]{l}x_{1},x_{2},\ldots,x_{n+1}\\ x_{1}^{\prime},x_{2}^{\prime},\ldots,x_{n+1}^{\prime}\end{array};f\right)=n!\mathrm{U}\left(x_{1}^{*},x_{2}^{*},\ldots,x_{n+1}^{*};f^{\prime}\right)

if $x_{i},x_{i}^{\prime}$ tend towards the point $x_{i}^{*}$ of $\mathrm{E}^{\prime},i=1,2,\ldots,n+1$
Since sequence (53) is ordered, from formulas (54) and (55) we deduce that if the function $f(x)$ is non-concave of order $n$ its derived nonconcave order $n-1$ The converse of this property is not true in general. The derivative $f^{\prime}(x)$ may be of order $n-1$ without the function being to cred $n$ it can be convex of order $n-1$ and the order function $n$ , without being convex. The ( $n-1$ The mth derivative of a function of order
^m is of order 1. Such a function has a left-hand derivative and a right-hand derivative, which are functions of order 0. If the $(n+1)^{\text{ème }}$ The derivative exists, and it has a constant sign.

If E is an interval and if the derivative $f^{\prime}(x)$ is non-concave of order $n-1$ the function $f(x)$ is necessarily non-concave of order $n$ Furthermore, if $f(x)$ is convex, $f^{\prime}(x)$ is also convex, and conversely, since one of these functions cannot be polynomial without the other also being polynomial in the same interval. In the case where E is an interval, if $f^{(n+1)}(x)$ exists, the condition $f^{(n+1)}(x)\geq 0$ is necessary and sufficient for the function to be non-concave of order $n$ the condition $f^{(n+1)}(x)>0$ is sufficient for it to be convex.

In the case of an interval, the properties of the function are simply deduced from those of the derivative by the easily established formula
(56) $\mathrm{U}\left(x_{1},x_{2},\ldots,x_{n+2};f\right)=n!\int_{x_{1}}^{x_{2}}\int_{x_{2}}^{x_{\mathrm{s}}}\ldots\int_{x_{n+1}}^{x_{n+2}}\mathrm{U}\left(t_{1},t_{2},\ldots,t_{n+1};f^{\prime}\right)dt_{1}dt_{2}\ldots dt_{n+1}$ 24.
We have seen that a function with bounded nth variation is a function with bounded nth difference divided by the function; therefore, a function with bounded nth variation has continuous derivatives of the same order. $1,2,\ldots,n-1$ If it is defined within a closed interval ( $a,b$ ) the continuous derivatives of order $r<n$ exist in the open interval at left.

We will also demonstrate that the derivative of a bounded nth variation function is at ( $n-1$ )th bounded variation.

Let's write the general formula (54) for the $2n$ points

\alpha_{1},\alpha_{1}^{\prime},\alpha_{2},\alpha_{2}^{\prime},\ldots,\alpha_{n},\alpha_{n}^{\prime}

Doing $f=x^{n}$ , We have

\sum_{1=1}^{n}\mathrm{\penalty 10000\ A}=\mathrm{U}\left(\begin{array}[]{l}\alpha_{1},\alpha_{2},\ldots,\alpha^{n}\\ \alpha_{1}^{\prime},\alpha_{2}^{\prime},\ldots,\alpha_{n}^{\prime}\end{array};x^{n}\right)

It is then obvious that if $\alpha_{1}{}^{*},\alpha_{2}{}^{*},\ldots,\alpha_{n}{}^{*}$ is an ordered sequence, at all $\varepsilon>0$ , corresponds to a $\eta>0$ such that one has

0<\frac{\mathrm{A}_{i}}{\mathrm{\penalty 10000\ V}\left(\alpha_{1}{}^{*},\alpha_{2}{}^{*},\ldots,\alpha_{n}{}^{*}\right)}<n!+\varepsilon,i=1,2,\ldots,n\left|\frac{\sum_{i=1}^{n}\mathrm{\penalty 10000\ A}_{i}}{\left.\mathrm{\penalty 10000\ V}\alpha_{1}{}^{*},\alpha_{2}{}^{*},\ldots,\alpha_{n}{}^{*}\right)}-n!\right|<\varepsilon

provided that $\left|\alpha_{1}-\alpha_{1}{}^{*}\right|<\eta,\quad\left|\alpha_{i}{}^{\prime}-\alpha_{i}{}^{*}\right|<\eta,\quad i=1,2,\ldots,n$ Now
consider an ordered sequence of $E^{\prime}$

x_{1},x_{2},\ldots x_{m}

(57)

According to the previous property, we can always take the ; points of E
(58)

y_{1},y_{2},\ldots,y_{2m}

so that, $\varepsilon>0$ given, we have

\begin{gathered}\left|\frac{\mathrm{U}\left(\begin{array}[]{l}y_{2i-1},y_{2i+1},\ldots,y_{2i+2n-3}\\ y_{2i},y_{2i+2},\ldots,y_{2i+2i-2}\end{array};f\right)}{\mathrm{V}\left(x_{i},x_{i+1},\ldots,x_{i+n-1}\right)}-\frac{\mathrm{U}\left(\begin{array}[]{l}y_{2i+1},y_{2i+3},\ldots,y_{2i+2n-1}\\ y_{2i+2},y_{2i+4},\ldots,y_{2i+2n}\end{array};f\right)}{\mathrm{V}\left(x_{i+1},x_{i+2},\ldots,x_{i+n}\right)}\right|<\\ <(2n-1)(n!+\varepsilon)\left\{\left|\Delta_{n}^{2i-1}-\Delta_{n}^{2i}\right|+\left|\Delta_{n}^{2i}-\Delta^{2i+1}\right|+\ldots\right.\\ \left.\cdots+\left|\Delta_{n}^{2i+n-1}-\Delta_{n}^{2i+n}\right|\right\}+2\varepsilon.\Delta_{n}[f;\mathrm{E}]\\ i=1,2,\ldots,m-n\end{gathered}

\Delta_{n}^{\prime}=\left[y_{i},y_{i+1},\ldots,y_{i+n};f\right],\quad j=1,2,\ldots,2m-n.

We deduce that
(59)

\begin{gathered}\sum_{i=1}^{m-n}\left\lvert\,\frac{\mathrm{U}\left(\begin{array}[]{l}y_{2i-1},y_{2i+1},\ldots,y_{2i+2n-3}\\ y_{2i},y_{2i+2},\ldots,y_{2i+2n-2}\end{array};f\right)}{\mathrm{V}\left(x_{i},x_{i+1},\ldots,x_{i+n-1}\right)}-\right.\\ \left.-\frac{\mathrm{U}\binom{y_{2i+1},y_{2i+3},\ldots,y_{2i+2n-1}}{y_{2i+2},y_{2i+4},\ldots,y_{2i+2n}}}{\mathrm{\penalty 10000\ V}\left(x_{i+1},x_{i+2},\ldots,x_{i+n}\right)}\right\rvert\,<\\ <(2n-1)(n!+\varepsilon)(n+1)\left\{\left|\Delta_{n}^{\prime}-\Delta_{n}^{2}\right|+\left|\Delta_{n}^{2}-\Delta_{n}^{3}\right|+\ldots\right.\\ \left.\cdots+4\Delta_{n}^{2m-n-1}-\Delta_{n}^{2m-n}\mid\right\}+2\varepsilon(m-n)\Delta_{n}[f;\mathrm{E}]\end{gathered}

But $\Delta_{n}[f;E]$ Having finished, with a slight change of notation we can also say that, given the sequence (57), we can determine the sequence (58) such that the left-hand side of (59) is

<(2n-1)\cdot(n+1)!V_{n}[f;E]+\frac{\varepsilon}{3}

$\varepsilon>0$ given in advance.
But the first member of (59) can be determined by a suitable choice of the sequence (57) so that it differs by less than $\frac{\varepsilon}{3}$ of: the quantity
(60)

\left(\begin{array}[]{ll}n&1\end{array}\right)!\cdot\left((n-1)\text{ ème variation de }f^{\prime}(x)\text{ sur }(57)\right)\text{. }

Finally, sequence (57) can be taken such that quantity (60) differs by less than $\frac{\varepsilon}{3(n-1)!}$ of $\left.V_{n-1}!^{\prime}f^{\prime}E^{\prime}\right]$ .

It follows that

\mathrm{V}_{n-1}\left[f^{\prime};\mathrm{E}^{\prime}\right]<(2n-1)(n+1)!\mathrm{V}_{n}[f;\mathrm{E}]+\varepsilon

done

\mathrm{V}_{n-1}\left[f^{\prime};\mathrm{E}^{\prime}\right]\leq(2n-1)(n+1)!\mathrm{V}_{n}[f;\mathrm{E}].

This inequality is rather crude, but sufficient to demonstrate the property. $f(x)$ being at a limited variation number $f^{(n-1)}(x)$ is therefore a first bounded variation of t. We follow that $f^{(n-1)}(x)$ then admits a left-hand derivative and a right-hand derivative which are of bounded variation of order $0\left({}^{37}\right)$ 25.
Let us first suppose that E is an interval ( $a,b$ We can give a precise meaning to the $(n+1)$ the divided difference of a function of order $n$ , even if the points are not all distinct.

It is about giving meaning to the divided difference

	$\displaystyle{[\underbrace{x_{1},x_{1},\ldots,x_{1}}_{i_{1}},\underbrace{x_{2},x_{2},\ldots,x_{2}}_{i_{2}},\underbrace{x_{k},x_{k},\ldots,x_{k}}_{i_{k}};f]}$		(61)
	$\displaystyle i_{1}+i_{2}+\cdots i_{k}=n+2$

the points $x_{1},x_{2},\ldots,x_{k}$ being distinct. If we consider (61) as the quotient of two determinants (see No. 1), it appears in indeterminate form. $\frac{0}{0}$ To resolve the uncertainty, we replace each group of points with the corresponding points. $x_{l},x_{j},\ldots,x_{j}$ by $i_{j}$ distinct points tending towards $x_{i}$ We can then obtain, using a well-known method (L'Hôpital's rule), the true value of the quotient (61). This amounts to defining the determinant

\mathbf{U}(\underbrace{x_{1},x_{1},\ldots,x_{1}}_{i_{1}},\underbrace{x_{2},x_{2},\ldots,x_{2}}_{i_{2}},\underbrace{x_{k},x_{k},\ldots,x_{k},}_{i_{k}};f)

as being equal to the determinant $\mathrm{U}\left(\alpha_{1},\alpha_{2},\ldots,\alpha_{n+2};f\right)$ where, generally speaking, the order lines are replaced $i_{1}+i_{2}+\ldots+i_{j-1}+1_{j}$ , $i_{1}+i_{2}+\cdots+i_{j-1}+2,\ldots,i_{1}+i_{2}+\cdots+i_{j-1}+i_{j}$ by :
$1x_{j}\quad a_{j}^{2}\quad x_{j}^{3}$ ………… $x_{j}^{n}\quad f\left(x_{j}\right)$
$\begin{array}[]{llll}0&1&2x_{j}&3x_{j}^{2}\end{array}nx_{j}^{n-1}f^{\prime}\left(x_{j}\right)$
$0\quad 0\quad 2\quad 6x_{j}$ . . $\_\_\_\_$
$\_\_\_\_$
$\_\_\_\_$
$\_\_\_\_$ .. $n(n-1)x_{j}^{n}$ 2 $\_\_\_\_$ $f^{\prime\prime}\left(x_{j}\right)\begin{array}[]{llll}0&0&0&0\end{array}$ 0 $\_\_\_\_$ . $\left(i_{j}-1\right)$ ! $n(n-1)$ $\_\_\_\_$ . $\left(n-i_{j}+2\right)x_{j}^{n}$ ).
$\mathrm{V}\left(x_{1},x_{1},\ldots,x_{1},x_{2},x_{2},\ldots,x_{2},\ldots,x_{k},x_{k},\ldots,x_{k}\right)$ is obtained by doing $f=x^{n+1}$ and we can easily see that this expression is different from zero, therefore the divided difference is perfectly defined.

The previous operation is justified if $k>2$ since the function has a continuous derivative of order $n-1$ The same applies if $k=2$ , $\mathrm{e}_{\mathrm{t}}2\leq i_{1}\leq n$ . If $k=2$ And $i_{1}=1$ We must introduce the nth derivative. This derivative does not exist in general, but there is always an nth left-hand derivative and an nth right-hand derivative. It follows that we can still give meaning to the divided difference, provided we let the $n+1$ points of the second group towards $x_{2}$ on the same side.

For example, if these points remain constantly to the left of $x_{2}$ the formula found is valid provided that $f^{(n)}\left(x_{2}\right)$ represents the nth left-hand derivative.
It is clear that these considerations are only valid within the interval ( $a,b$ ), more precisely, everywhere derivatives exist.

We can now complete the properties of functions of order n. If $f(x)$ is non-concave of order $n$ we have

\left[x_{1},x_{2},\ldots,x_{n+2};f\right]\geq 0

the points being distinct or not. We will also see that if $f(x)$ is of order $n$ and if we have

\begin{gathered}{\left[x_{1},x_{2},\ldots,x_{n+2};t\right]=0}\\ x_{1}\leq x_{2}\leq\cdots\leq x_{n+2}\quad(\text{ non pas tous confondus })\end{gathered}

The function is a polynomial of order $n$ in the meantime ( $x_{1},x_{n+2}$
Moreover, all these properties can be translated geometrically using L polynomials .

If the set E is arbitrary, the preceding considerations still apply, provided that the points where several coincide belong to a derived set of suitable order. The derivatives can also be extended by the expression $n!f_{n}(x)$ across the entire set $\mathrm{E}^{\prime}$ It is easy to verify that this extension allows for the complete generalization of the properties. For example, if $f(x)$ is non-concave of order $n$ It is shown that everywhere they exist $f_{1}(x),f_{2}(x),\ldots$ are also non-concave in order $n-1,n-2,\ldots$ 26 respectively
. Suppose that $x_{1},x_{2},\ldots,x_{n}$ let be points of the derived set. We can take $x_{1}{}^{\prime},x_{2}{}^{\prime},\ldots,x_{n}{}^{\prime}$ sufficiently close to the respective points of the preceding sequence such that, $\varepsilon>0$ given, we have
with formula (54)

\left|\frac{\mathrm{U}\binom{x_{1},x_{2},\ldots,x_{n}}{x_{1}^{\prime}\cdot x_{2}^{\prime},\ldots,x_{n}^{\prime};f}}{\mathrm{\penalty 10000\ V}\left(x_{1},x_{2},\ldots,x_{n}\right)}-(n-1)!\left[x_{1},x_{2},\ldots,x_{n};f^{\prime}\right]\right|<\frac{\varepsilon}{2}(n-1)!

\left|\frac{\sum_{i=1}^{n}\mathrm{\penalty 10000\ A}_{i}}{\mathrm{\penalty 10000\ V}\left(x_{1},x_{2},\ldots,x_{n}\right)}-n!\right|<\frac{\varepsilon}{2J_{n}}(n-1)!

assuming the function has a bounded nth difference divided function.
We deduce

\left|\mid x_{1},x_{2},\ldots,x_{n};f^{\prime}\right]\mid<n.\Delta_{n}[f;\mathbb{E}]+\varepsilon

from where

\Delta_{n-1}\left[f^{\prime};\mathrm{E}^{\prime}\right]\leq n\Delta_{n}[t;\mathrm{E}]

(62)

When the set E is an interval ( $a,b$ ) Formula (56) allows us to write (for $n+1$ points)

\mathrm{U}\left(x_{1},x_{2},\ldots,x_{n+1};f\right)=

$=(n-1)!\int_{x_{1}}^{x_{2}}\int_{x_{3}}^{x_{3}}\cdots\int_{x_{n}}^{x_{n+1}}\left[t_{1},t_{2},\ldots,t_{n};f^{\prime}\right],\mathrm{V}\left(t_{1},t_{2},\ldots,t_{n}\right)dt_{1}dt_{2}\ldots dt_{n}\ldots$
We also have
$\mathrm{V}\left(x_{1},x_{2},\ldots,x_{n+1}\right)=n!\int_{x_{1}}^{x_{2}}\int_{x_{2}}^{x_{3}}\ldots\int_{x_{n}}^{x_{n+1}}\mathrm{\penalty 10000\ V}\left(t_{1},t_{2},\ldots,t_{n}\right)dt_{1}dt_{2}\ldots dt_{n}\ldots$
We deduce from this

n.\left|\left[x_{1},x_{2},\ldots,x_{n+1};f\right]\right|\leq\Delta_{n-1}\left[f_{a}^{b}\right]

from where

\left.n\Delta_{n}[\underset{a}{b}]\leq\Delta_{n}\stackrel{{\scriptstyle b}}\frac{b}{a}\right]

Comparing with (62) we deduce equality for the case of an interval.

\Delta_{n-1}\left[f_{a}^{b}\right]=n\Delta_{n}\left[f_{a}^{b}\left({}^{38}\right)\right.

(63)

27.

We already know that if $f(x)$ is of bounded nth variation $f^{\prime}(x)$ is at ( $n-1$ )th bounded variation. We also know that if
⁽³⁸⁾ It can easily be verified that this equality does not hold in general if the set E is arbitrary. This is, moreover, obvious a priori since: the differentiation does not exist on isolated points of E.
function is continuous (which always holds if $n\geq 1$ ) we can consider only equidistant points for the study of variation. Suppose that E is an interval $(a,b)$ and that $n>1$
To all $\varepsilon>0$ corresponds to a sequence

64.

$x,x+h,x+2h,\ldots,x+mh,\quad(h>0)$
such that
(665) (nth variation of $f$ on (64) $>V_{n}\left[\frac{b}{a}\right]-\varepsilon$ .

We can always assume (as a result of continuity) $a<x$ , $x+mh<b$ .

It is easy to see that

[x+ih,x+(i+1)h,\ldots,x+(n+i)h;f]=

=\frac{1}{n}\int_{0}^{1}\left[ht+x+ih,ht+x+(i+1)h,\ldots,ht+x+(n+i-1)h;t^{\prime}\right]dt

"and comparing with (65) it follows that

n\left(\mathrm{\penalty 10000\ V}_{n}[\stackrel{{\scriptstyle b}}]-\varepsilon\right)<\mathrm{V}_{n-1}\left[\stackrel{{\scriptstyle b}}_{a}^{b}\right]

from where

n\mathrm{\penalty 10000\ V}_{n}[\stackrel{{\scriptstyle b}}]\leq\mathrm{V}_{n-1}[\stackrel{{\scriptstyle b}}{{\underset{a}{f}}}]

Now, at all $\varepsilon>0$ corresponds to a number $\eta>0$ such as $\left.\mid\left\{\mid x+y,x+2y,\ldots,x+ny;t^{\prime}\right]-\left[x,x+y,\ldots x+(n-1)y;t^{\prime}\right]\right\}-$
$\left.-\frac{ny}{h}\{[x,x+y,\ldots,x+ny;f]-[x+h,x+y+h,\ldots,x+ny+h;f]\}\right\rvert\,<\varepsilon$ Provided that $|h|<\eta$ .

Using formulas (6) and (10) we can write
(66) $[x+h,x+y+h,\ldots,x+ny+h;f]-[x,x+y,\ldots,x+ny;f]=$

\begin{gathered}=\mathrm{A}_{1}[x,x+h,x+y,\ldots;f]+\mathrm{A}_{2}[x+h,x+y,x+y+h,\ldots;f]+\ldots\\ \ldots+\mathrm{A}_{n+1}[\ldots,x+(n-1)y+h,x+ny,x+ny+h;f]\end{gathered}

all the differences divided by the second member being of order $n+1$ ules $\mathrm{A}_{l}$ are independent of $f(x)$ and also of $x$ and they are non-mogative if $x<x+h<x+y$ .

We can find a sequel

x,x+y,\ldots,x+my,\quad y>0,a<x,x+my<b

such as, $\varepsilon>0$ given, we have

<\begin{gathered}\mathrm{V}_{n+1}\left[f_{a}^{b}\right]-\frac{\varepsilon}{2}<\\ \left.<\sum_{\imath=1}^{m-n+1}\mid x+iy,x+(i+1)y,\ldots,x+(n+i-1)y;f^{\prime}\right]-\\ -\left[x+(i-1)y,x+iy,\ldots,x+(n+i-2)y;f^{\prime}\right]\mid.\end{gathered}

We take $h$ small enough $x<x+h<x+y,x+my+h<b$ ,
And

\begin{gathered}\sum_{\imath=1}^{m-n+1}\mid\left[x+iy,x+(i+1)y,\ldots,x+(n+i-1)y;f^{\prime}\right]-\\ -\left[x+(i-1),x+iy,\ldots,x+(n+i-2)y;f^{\prime}\right]\mid<\\ \left.<\frac{\varepsilon}{2}+\frac{ny}{h}\sum_{\imath=1}^{m-n+1}\right\rvert\,[x+(i-1)y+h,x+iy+h,\ldots,x+(n+i-1)y+h;f]-\\ {[x+(i-1)y,x+iy,\ldots,x+(n+i-1)y;f]\mid.}\end{gathered}

However, formula (66) shows that the sum of the right-hand side is less than

\mathrm{A}^{*}\mathrm{\penalty 10000\ V}_{n}[\stackrel{{\scriptstyle b}}]

Or
$A^{*}=\frac{2}{(n+1)y}\cdot\max.\left(A_{1}+A_{3}+A_{5}+,\ldots,A_{2}+A_{4}+A_{6}+\ldots\right)n$ odd $=\max.\left(\frac{\mathrm{A}_{1}+\mathrm{A}_{3}+\mathrm{A}_{5}+\ldots}{\frac{n}{2}y+h},\frac{\mathrm{\penalty 10000\ A}_{2}+\mathrm{A}_{4}+\mathrm{A}_{6}+\ldots}{\left(\frac{n}{2}+1\right)y-h}\right)\quad n$ even.
If $n$ is odd, you just need to do $f=x^{n+1},x^{n+2}$ to see that

\mathrm{A}^{*}=\frac{h}{y}.

Relation (1) then allows us to show that this equality also holds for $n$ peer.

We deduce from this

\mathrm{V}_{n-1}[\underset{a}{f}]-\varepsilon<n\mathrm{\penalty 10000\ V}_{n}[\underset{a}{b}]

hence

\mathrm{V}_{n-1}[\underset{a}{\stackrel{{\scriptstyle b}}}]\leq n\mathrm{\penalty 10000\ V}_{n}[\underset{a}{b}].

So, finally we have
(67)

\mathrm{V}_{n-1}\left[{\underset{a}{f}}^{b}\right]=n\mathrm{\penalty 10000\ V}_{n}[\underset{a}{b}.

We assumed that $t^{\prime}$ either continuous. MA Winternitz has shown that we also have ( ³⁹ )

\mathrm{V}_{0}\left[{}_{a}^{b}\right]=\mathrm{V}_{1}\left[{}_{a}^{b}\right]

$f$ being of bounded first variation and $f_{d}{}^{\prime}$ its right-hand derivative.

§ 3. - On the limitation of the derivative of a function of order $n$ .

28.

Either $f(x)$ non-concave of order $n$ in the meantime ( $a,b$ ). If $n>1$ it admits a continuous, therefore bounded, derivative in every completely interior interval.

The sequel $x_{1},x_{2},\ldots,x_{k},x,x^{\prime},x_{k+1},\ldots,x_{n}$ being ordered, we have

\lim_{x^{\prime}\rightarrow x}\frac{\mathrm{U}\left(x_{1},x_{2},\ldots,x_{k},x,x^{\prime},x_{k+1},\ldots,x_{n};f\right)}{x^{\prime}-x}\geq 0

hence, by expanding
(68) $(-1)^{n-k-1}f^{\prime}(x)$

\left.=||\begin{array}[]{cccccc}1&x_{1}&x_{1}^{2}&\ldots\ldots&x_{1}^{n}&f\left(x_{1}\right)\\ 1&x_{2}&x_{2}^{2}&\ldots\ldots&x_{2}^{n}&f\left(x_{2}\right)\\ \ldots\ldots\ldots&\ldots\ldots&\ldots&\ldots\\ 1&x_{k}&x_{k}^{2}&\ldots\ldots&x_{k}^{n}&f\left(x_{k}\right)\\ 1&x&x^{2}&\ldots\ldots&x^{n}&f(x)\\ 0&4&2x&\ldots\ldots&x_{n}^{n+1}&0\\ 1&x_{k11}&x_{k+1}^{2}&\ldots\ldots&x_{k}^{n}&f\left(x_{k+1}\right)\\ \cdots&\ldots&\ldots&\ldots&\ldots&\ldots\\ 1&x_{n}&x_{n}^{2}&\ldots&\ldots&x_{n}^{n}\end{array}\quad f\left(x_{n}\right)\quad\right\rvert\,

To find a limitation of $f^{\prime}$ in the meantime ( $a_{1},b_{1}$ ) $a<a_{1}<b_{1}<b$ you just need to take the points $x_{1},x_{2},\ldots,x_{k}$ In
( $a,a_{1}$ ) the points $x_{k+1},x_{k+2},\ldots,x_{n}$ In ( $b_{1},b$ and to determine appropriately $k$ (one can always choose between two consecutive values of $k)$ so that $(-1)^{n-k-1}f^{\prime}(x)>0$ We can then take the modules in (68).

Either $\Delta_{0}$ the terminal of $f(x)$ and let's ask

\mathrm{P}(z)=(z-x)\left(z-x_{1}\right)\left(z-x_{2}\right)\ldots\left(z-x_{n}\right).

The second member of (68) can be written

\left|\mathrm{P}^{\prime}(x)\right|\sum_{i=1}^{n}\frac{(-1)^{n-t}f\left(x_{i}\right)}{\left|x-x_{i}\right|.\left|\mathrm{P}^{\prime}\left(x_{i}\right)\right|}+(-1)^{n-k+1}f(x)\sum_{i=1}^{n}\frac{1}{x-x_{i}}

and this expression allows us to write

\left|f^{\prime}\right|\leq\Delta_{0}\left\{\sum_{i=1}^{n}\frac{\left|\mathrm{P}^{\prime}(x)\right|}{\left|x-x_{i}\right|\cdot\left|\mathrm{P}^{\prime}\left(x_{i}\right)\right|}+\sum_{i=1}^{n}\frac{1}{\left|x-x_{i}\right|}\right\}

$k$ being suitably chosen.
Let's look for a better limitation for sufficiently internal points.

We will assume that $f(x)=0$ Let's then ask...

\mathrm{B}_{k}=\left|\mathrm{P}^{\prime}(x)\right|\sum_{i=1}^{n}\frac{1}{\left|x-x_{i}\right|\cdot\left|\mathrm{P}^{\prime}\left(x_{i}\right)\right|}=(-1)^{n-k}\mathrm{P}^{\prime}(x)\sum_{i=1}^{n}\frac{(-1)^{n-i+1}}{\left(x-x_{i}\right)\mathrm{P}^{\prime}\left(x_{i}\right)}

And
$\mathrm{B}^{*}{}_{k}=\min$ . of $\mathrm{B}_{k},\mathrm{\penalty 10000\ B}^{*}{}_{i,j}=\max.\left(\mathrm{B}^{*}{}_{i},\mathrm{\penalty 10000\ B}^{*}{}_{j}\right)\quad i+j=$ odd.
We deduce that
$\left|f^{\prime}\right|\leq\Delta_{0}.\min.\left(\mathrm{B}^{*}{}_{i,j}\right)\quad i,j=0,1,2,\ldots,n,\quad i+j=$ odd.
Let us also designate by $F(z)$ the polynomial $L$ taking the values $(-1)^{n-i+1}$ to the points $x_{i},i=1,2,\ldots,n+1\left(x_{n+1}=x\right)$ , We have

		$\displaystyle\frac{\mathrm{F}(z)}{\mathrm{P}(z)}=\sum_{\imath=1}^{n+1}\frac{(-1)^{n-i+1}}{\left(z-x_{i}\right)\mathrm{P}^{\prime}\left(x_{i}\right)}\quad\left(x_{n+1}=x\right)$
		$\displaystyle\mathrm{B}_{k}=(-1)^{n-k}\left(\mathrm{\penalty 10000\ F}^{\prime}(x)-\frac{\mathrm{P}^{\prime\prime}(x)}{2\mathrm{P}^{\prime}(x)}\right)$
		ment if we ask

	$\displaystyle\mathrm{P}(z)$	$\displaystyle=(z-x)\left(z-x_{1}\right)\left(z-x_{n}\right)Q(z)$
	$\displaystyle\mathrm{R}(z)$	$\displaystyle=\mathrm{F}(z)-\frac{\left(z-x_{1}\right)\left(z-x_{n}\right)Q(z)}{\left(x-x_{1}\right)\left(x-x_{n}\right)Q(x)}$

we deduce from this

\mathrm{B}_{k}=(-1)^{n-k}\mathrm{R}^{\prime}(x)=\left|\mathrm{R}^{\prime}(x)\right|

29.

We can see that if $x$ is fixed $\mathrm{B}_{k}$ is positive and never cancels out. If two points $x_{i}$ And $x$ coincide $\mathrm{B}_{k}$ becomes infinite, so it certainly reaches its minimum for distinct points. Finally $\mathbf{B}_{k}$ is homogeneous of degree -1 in $x_{i}$ And $x$ and depends only on the mutual differences of these numbers. It follows that for the minimum, at least one of the equalities $x_{1}=a,x_{n}=b$ , must be verified.

Suppose $x_{1}=a$ and let's write the conditions

\frac{\partial\mathrm{B}_{k}}{\partial x_{j}}=0,\quad j=2,3,\ldots,n

The value thus found for $\mathrm{B}_{k}$ will be $\geq\mathrm{B}_{k}^{*}$ .
We have

\frac{\partial\mathrm{B}_{k}}{\partial x_{j}}=\frac{(-1)^{n-k}\mathrm{P}^{\prime}(x)}{x_{j}-x}\left(\frac{\mathrm{\penalty 10000\ F}^{\prime}\left(x_{j}\right)}{\mathrm{P}^{\prime}\left(x_{j}\right)}-\frac{1}{\left(x_{j}-x\right)\mathrm{P}^{\prime}(x)}\right)

\left(x_{j}-x\right)\mathrm{P}^{\prime}(x)\mathrm{F}^{\prime}\left(x_{j}\right)=\mathrm{P}^{\prime}\left(x_{j}\right)\quad j=2,3,\ldots,n

assuming the points $x_{f},x$ distinct.
We finally deduce
$\mathrm{R}(x)=0,\mathrm{R}\left(x_{i}\right)=(-1)^{n-i+1},i=1,2,\ldots,n,\mathrm{R}^{\prime}\left(x_{i}\right)=0,i=2,3,\ldots,n$ To
highlight the number $k$ let us denote this polynomial $\operatorname{par}\mathrm{T}_{k}(z)$ .

Now suppose that $x_{n}=b$ and let's write

\frac{\partial\mathrm{B}_{k}}{\partial x_{j}}=0,\quad j=1,2,\ldots,n-1

we then obtain for $\mathrm{B}_{k}$ a value $\geq\mathrm{B}^{*}{}_{k}$ As above, we find
$\mathrm{R}(x)=0,\mathrm{R}\left(x_{i}\right)=(-1)^{n-i+1},i=1,2,\ldots,n,\mathrm{R}^{\prime}\left(x_{i}\right)=0,i=1,2,\ldots,n-1$
Let us denote this polynomial by $-\overline{\mathrm{T}}_{k}(z)$ .
We have

\begin{array}[]{ll}\mathrm{B}_{k}^{*}\leq\left|\mathrm{T}^{\prime}(x)\right|&\text{ pourvu que }x_{n}\leq b\\ \mathrm{\penalty 10000\ B}_{k}^{*}\leq\left|\overline{\mathrm{T}}^{\prime}(x)\right|&\text{ pourvu que }x_{1}\geq a.\end{array}

It can also be demonstrated that equality is indeed achieved for a particular position of the point $x$ .

The polynomial $\mathrm{T}(z)$ is a Chebyshev polynomial ( ⁴⁰ ) in a

⁰⁰footnotetext: (40) Voir S. Beinstein loc. cit. (8).

a certain interval ( $a,b$ , )

\mathrm{T}_{k}(z)=\cos\left[n\arccos\frac{2z-b_{1}-a}{b_{1}-a}\right]

For formula (69) to be applicable, it is necessary to write that $x$ is the Kth root of $\mathrm{T}_{k}(z)=0$ and that $x_{n}$ , which is the largest root of the derivative, is at most equal to $b$ ,
$x=\frac{b_{1}+a}{2}-\frac{b_{1}-a}{2}\cos\frac{2k-1}{2n}\pi,\quad b\geq x_{n}=\frac{b_{1}+a}{2}+\frac{b_{1}-a}{2}\cos\frac{\pi}{n}$ Eliminating
$b_{4}$ we find

x\leq\dot{\xi}_{k}=\frac{a\left(\cos\frac{\pi}{n}+\cos\frac{2k-1}{2n}\pi\right)+b\left(1-\cos\frac{2k-1}{2n}\pi\right)}{1+\cos\frac{\pi}{n}}

The points $\xi_{k}$ are distributed as follows:

a<\xi_{1}<\xi_{2}<\ldots<\xi_{n-1}<b<\xi_{n}

In the same way we have

\bar{T}_{k}(z)=\cos\left[n\arccos\frac{2z-a_{1}-b}{b-a_{1}}\right]

the conditions for the applicability of formula (70) are

x=\frac{b+a_{1}}{2}+\frac{b-a_{1}}{2}\cos\frac{2k+1}{2n}\pi,\quad a\leq x_{1}=\frac{b+a_{1}}{2}+\frac{b-a_{1}}{2}\cos\frac{\pi}{n}

from where

x\geq\eta_{k}=\frac{a\left(1+\cos\frac{2k+1}{2n}\pi\right)+b\left(\cos\frac{\pi}{n}-\cos\frac{2k+1}{2n}\pi\right)}{1+\cos\frac{\pi}{n}}

The paintings $\eta_{k}$ are distributed as follows:

\eta_{0}<a<\eta_{1}<\eta_{2}<\ldots<\eta_{n-1}<b.

30.

If $x$ is in the interval $\left(\xi_{k-1},\xi_{k}\right)$ polynomials $\mathrm{T}_{m}(z)$ , $m=k,k+1,\ldots$ suitable for limitation. We have

\left|T_{m}^{\prime}(x)\right|=\frac{n}{\sqrt{(x--a)\left(b_{1}-x\right)}}=\frac{n}{x-a}\operatorname{tg}\frac{2n-1}{4n}\pi

\left|\mathrm{T}^{\prime}{}_{k}(x)\right|<\left|\mathrm{T}_{k+1}^{\prime}(x)\right|<\left|\mathrm{T}_{k+2}^{\prime}(x)\right|<\ldots

and we will then certainly be able to write

\left|f^{\prime}(x)\right|\leq\Delta_{0}.\left|\mathrm{T}_{k+1}^{\prime}(x)\right|\quad\quad\text{ dans }\left(\xi_{k-1},\xi_{k}\right)^{)}

Similarly, we find

\left|f^{\prime}(x)\right|\leq\Delta_{0}\cdot\left|\overline{\mathrm{\penalty 10000\ T}}_{k-1}^{\prime}(x)\right|\quad\text{ dans }\left(\eta_{k},\eta_{k+1}\right)

with

\left|\overline{\mathrm{T}}_{k-1}^{\prime}(x)\right|=\frac{n}{\sqrt{\left(x-a_{1}\right)(b-x)}}=\frac{n}{b-x}\operatorname{cotg}\frac{2k-1}{4n}\pi

We have in $\left(\xi_{k-1},\xi_{k}\right)$

\begin{gathered}\left|\mathrm{T}^{\prime}k+1(x)\right|=\frac{n}{x-a}\operatorname{tg}\frac{2k+1}{4n}\pi\leq\frac{n}{5k-1-a}\operatorname{tg}\frac{2k+1}{4n}\pi=\\ =\frac{n}{b-a}\frac{1+\cos\frac{\pi}{n}}{1-\cos\frac{2k-3}{2n}\pi}\cdot\operatorname{tg}\frac{2k+1}{2n}\pi=\frac{n}{b-a}\frac{1+\cos\frac{\pi}{n}}{\sin\frac{2k-3}{2n}}\cdot\frac{\operatorname{tg}\frac{2k+1}{4n}\pi}{\operatorname{tg}\frac{2k-3}{4n}\pi}<\\ <\frac{2n}{b-a}\frac{2n}{\pi}\frac{\frac{\pi}{2n}}{\sin\frac{\pi}{2n}}\frac{\operatorname{tg}\frac{2k+1}{4n}\pi}{\operatorname{tg}\frac{2k-3}{4n}\pi}<\frac{4n}{b-a}\frac{1}{\pi}\frac{\frac{\pi}{6}}{\sin\frac{\pi}{6}}\frac{\operatorname{tg}\frac{2k+1}{4n}\pi}{\operatorname{tg}\frac{2k-3}{4n}\pi}\end{gathered}

and finally

\begin{gathered}\left|\mathrm{T}_{k+1}^{\prime}(x)\right|<\frac{4}{3}\frac{n^{2}}{b-a}\cdot\frac{\operatorname{tg}\frac{2k+1}{4n}\pi}{\operatorname{tg}\frac{2k-3}{4n}\pi}\quad\text{ dans }\left(\xi_{k-1},\xi_{k}\right)\mid\\ n>2,k=2,3,\ldots,n-1\end{gathered}

In the same way we have

\begin{gathered}\left|\overline{\mathrm{T}}_{k-1}^{\prime}(x)\right|<\frac{4}{3}\frac{n^{2}}{b-a}\frac{\operatorname{tg}\frac{2k+3}{4n}\pi}{\operatorname{tg}\frac{2k-1}{4n}\pi}\quad\text{ dans }\left(\eta_{k},\eta_{k+1}\right)\\ n>2,k=1,2,\ldots,n-2\end{gathered}

We can also write in $\left(\xi_{k-1},\xi_{k}\right)$

	$\displaystyle\\|\mathrm{T}_{k+1}^{*}(x)\mid$	$\displaystyle=\frac{n}{\sqrt{(x-a)(b-x)}}=\frac{n}{\sqrt{(x-a)(b-x)}}\sqrt{\frac{b-x}{x-a}}\cdot\operatorname{tg}\frac{2k-1-1}{4n}\pi\leq$
		$\displaystyle\leq\frac{n}{\sqrt{(x-a)(b-x)}}\sqrt{\frac{b-\xi_{k-1}}{\xi_{k-1}-a}\operatorname{tg}\frac{2k+1}{4n}\pi=}$
		$\displaystyle=\frac{n}{\sqrt{(x-a)(b-x)}}\cdot\frac{\operatorname{tg}\frac{2k+1}{4n}\pi}{\operatorname{tg}\frac{2k-3}{4n}\pi}\sqrt{\frac{\cos\frac{2k-5}{4n}\pi\cdot\cos\frac{2k-1}{4n}\pi}{\cos^{2}\frac{2k-3}{4n}\pi}}$

from where finally

\begin{gathered}\left|\mathrm{T}_{k+1}^{\prime}(x)\right|<\frac{n}{\sqrt{(x-a)(b-x)}}=\frac{\operatorname{tg}\frac{2k+1}{4n}\pi}{\operatorname{tg}\frac{2k-3}{4n}\pi}\\ n>2,k=2,3,\ldots,n-1\end{gathered}\quad\text{ dans }\left(\xi_{k-1},\xi_{k}\right)

In a similar way

\begin{gathered}\left|\bar{T}_{k-1}^{\prime}(x)\right|<\frac{n}{\sqrt{(x-a)(b-x)}}\cdot\frac{\operatorname{tg}\frac{2k+3}{4n}\pi}{\operatorname{tg}\frac{2k-1}{4n}\pi}\\ n>2,k=1,2,\ldots,n-2.\end{gathered}\text{ dans }\left(\eta_{k},\eta_{k+1}\right)

Note that

\frac{\operatorname{tg}\frac{2k+1}{4n}\pi}{\operatorname{tg}\frac{2k-3}{4n}\pi}\leq\frac{\operatorname{tg}\frac{5\pi}{4n}}{\operatorname{tg}\frac{\pi}{4n}}\leq\frac{\operatorname{tg}\frac{5\pi}{12}}{\operatorname{tg}\frac{\pi}{12}}=\frac{2+\sqrt{3}}{2-\sqrt{3}},\quad n>2,k=2,3,\ldots,n-1

And

\begin{gathered}\eta_{1}>\xi_{1}=\frac{a\left(\cos\frac{\pi}{n}+\cos\frac{\pi}{2n}\right)+b\left(1-\cos\frac{\pi}{2n}\right)}{1+\cos\frac{\pi}{n}}\\ \xi_{n-1}<\eta_{n-1}=\frac{a\left(1-\cos\frac{\pi}{2n}\right)+b\left(\cos\frac{\pi}{n}+\cos\frac{\pi}{2n}\right)}{1+\cos\frac{\pi}{n}}\end{gathered}

We can therefore see that:
The derivatives of a function $f(x)$ bounded and orderly $n$ in the intex-
valley ( $a,b$ ) checks the inequalities ( $n>2$ )

\left\{\begin{array}[]{l}\left|f^{\prime}(x)\right|<\frac{8}{3}\cdot\frac{2+\sqrt{3}}{2-\sqrt{3}}\cdot\frac{n^{2}}{b-a}\Delta_{0}\\ \left|f^{\prime}(x)\sqrt{(x-a)(b-x)}\right|<2\frac{2+\sqrt{3}}{2-\sqrt{3}}n\cdot\Delta_{0}\end{array}\right.

$x$ belonging to the inter-flight ( $a+\lambda,b-\lambda$ ) with

\lambda=(b-a)\frac{1-\cos\frac{\pi}{2n}}{1+\cos\frac{\pi}{n}}

We have introduced the coefficient 2 here; it comes from the fact that, in the proof, we have assumed $f(x)=0$ The general case reduces to this one by considering the function $f_{1}(z)=f(z)-f(x)$ which is still of order $n$ and we obviously have

f_{1}(x)=0,f_{1}^{\prime}(z)=f^{\prime}(z),\left|f_{1}(z)\right|\leq 2\Delta_{0}.

MP Montel was kind enough to point out to us that, in the vicinity of the extremities $a+\lambda$ Or $b-\lambda$ the second limitation is comparable to the first for large values of $n$ It suffices to note that $\lambda$ is of the order of $\frac{1}{n^{2}}$ .

Formulas (71) closely resemble those given by Markoff and M.S. Bernstein for the limitation of the derivative of a polynomial ( ⁴¹ ). It can be seen that a function of order $n$ behaves roughly like a polynomial of degree n in a subinterval, which, moreover, can be approached as closely as one wants to $(a,b)$ For $n\rightarrow\infty$ .

Chapter IV.

ON CONVEX FUNCTIONS IN THE SENSE OF M. JENSEN.
31. In his famous memoir MJ Jensen ( ⁴² ) defines convex (ordinary) functions by considering only second divided differences taken on equidistant points.
$\left({}^{41}\right)$ See S. Bernstein loc. cit. $\left({}^{8}\right)$ ( 42
) JLWV Jensen, "On Convex Functions and Inequalities Between Mean Values," Acta Math., vol. 30 (1906), p. 175. M. L. Galvani was the first to consider functions defined on arbitrary sets. See his memoir "Sulle funzioni convesse di una o due variabili definite in un aggregato qualunque," Rendiconti di Palermo, vol. 41 (1916), p. 103.

Let us assume, for the sake of clarity, that E is an interval ( $a,b$ ) and consider the divided differences

\delta_{n}(x;h;f)=\frac{1}{n!h^{n}}\sum_{i=0}^{n}\binom{n}{i}f(x+(n-i)h).

Let's suppose that $f(x)$ let be bounded and let us set

\begin{gathered}\lim.\sup.\left|\sum_{i=1}^{n}(-1)^{t}\binom{n}{i}f(x+(n-i)h)\right|=\omega_{n}(\delta)\\ a\leq x,x+h\leq b,\quad|h|\leq\delta\end{gathered}

$\omega_{n}(\delta)$ is the oscillation modulus of order $n$ of the function.
Let's

\Delta_{n}^{\prime}=\lim.\sup.\left|\delta_{n}(x;h;f)\right|\quad\operatorname{dans}(a,b).

We have $\omega_{n}(\delta)\leq n!\delta^{n}\Delta_{n}^{\prime}$ , so if $\Delta_{n}^{\prime}$ is finished

\lim_{\delta\rightarrow 0}\omega_{n}(\delta)=\dot{0}

MA Marchaud has shown that in this case the function is continuous ( ⁴³ ).

Let's take $n+1$ ordered points $x_{1},x_{2},\ldots,x_{n+1}$ and divide the interval ( $\omega_{1},x_{n+1}$ ) in $p$ equal parts by points

x_{1}=x_{0}^{\prime},x_{1}^{\prime},\ldots,x_{p-1}^{\prime},x_{p}^{\prime}=x_{n+1}

Either $x_{i}{}^{\prime\prime}$ the point $x_{j}{}^{\prime}$ which is closest to $x_{i}$ (or one of them if there are two). Formula (10) then gives us

\left|\left[x_{1}^{\prime\prime},\ldots,x_{n+1}^{\prime\prime};f\right]\right|\leq\Delta_{n}^{\prime}.

But

\left|x_{i}-x^{\prime\prime}\right|\leq\frac{x_{n+1}-x_{1}}{2p}

and since the function is continuous, we can find a number $\eta>0$ such as

\left|\left[x_{1},x_{2},\ldots,x_{n+1};f\right]-\left[x^{\prime\prime}{}_{1},x^{\prime\prime}{}_{2},\ldots,x^{\prime\prime}{}_{n+1};f\right]\right|<\varepsilon

$\varepsilon>0$ being any given number, provided that $p>\eta$ . It follows that

If $\Delta^{\prime}{}_{n}$ is finite and if the function is bounded it is also a bounded nth difference divided function and we have

\Delta_{n}=\Delta_{n}^{\prime}.

(43) A. Marghaud loc, cit. (7). .

We can also define a total variation $V^{\prime}{}_{n}$ on equidistant points by setting

	$\displaystyle\mathrm{V}_{n}^{\prime}$	$\displaystyle=\lim.\sup.\sum_{i=0}^{m}\left\|\delta_{n}(x+ih;h;f)-\delta_{n}(x+(i+1)h;h;f)\right\|=$
		$\displaystyle=\lim.\sup.(n+1)h\sum_{\imath=1}^{m}\left\|\delta_{n}{}_{1}(x+ih;h;f)\right\|\quad(h>0)$

If $\Delta^{\prime}{}_{n+1}$ is limited, the same is true for $\mathrm{V}^{\prime}{}_{n}$ , but $\mathrm{V}^{\prime}{}_{n}$ can be limited without $\Delta_{n}^{\prime}$ the evening.

It is further demonstrated, as above, that if $\mathrm{V}^{\prime}{}_{n}$ is finished and the function $f(x)$ bounded, it has a bounded nth variation and we have

V_{n}=V_{n}^{\prime}.

Of course, this property is only true for $n>0$ We generally have $V_{0}>V_{0}^{\prime}$ 32.
We can finally consider functions $f(x)$ verifying the inequality

\delta_{n+1}(x;h;f)\geq 0\quad\text{ dans }(a,b)

(72)

I say first that if such a function is bounded it is continuous in the open interval ( $a,b$ ) ( $n>0$ ).

From formula (10) it follows that we have

\left[x_{1},x_{2},\ldots,x_{n+2};f\right]\geq 0

(73)

provided that the points $x_{2},x_{3},\ldots,x_{n+1}$ rationally divide the interval ( $x_{1},x_{n+2}$ ). So then $x$ an interior point of ( $a,b$ ) and let's $n$ fixed points $x_{1},x_{2},\ldots,x_{n}$ ordered and to the left of $x$ and either $x^{\prime}$ a point near $x$ , which we can assume to be to the right of $x$ To clarify things. We have

\mathrm{U}\left(x_{1},x_{2},\ldots,x_{n},x,x^{\prime};f\right)\geq 0

(74)

pourvu que la condition de rationalité soit vérifiée. Or, nous pouvons toujours prendre les points $x_{i}$ de manière que $x_{2},x_{3},\ldots,x_{n}$ divisent rationnellement l’intervalle ( $x_{1},x$ ). Alors, ou bien $x_{2},x_{3},\ldots,x_{n},x$ divisent rationnellement l’intervalle ( $x_{1},x^{\prime}$ ), ou bien nous pouvons rem-
placer les points $x_{i}$ par d’autres $x_{i}^{\prime}$ aussi près qu’on veut des $x_{i}$ tels que cette propriété soit vérifiee. En développant alors l’inégalité (74) nous avons

f\left(x^{\prime}\right)-f(x)\geq\left(x^{\prime}-x\right)\frac{\mathrm{A}}{\mathrm{\penalty 10000\ V}\left(x_{1},x_{2},\ldots,x_{n},x\right)}=\left(x^{\prime}-x\right)\mathrm{A}_{1}.

(75)

La quantité A est bornée la fonction l’étant aussi par hypothèse. Dans $\mathrm{V}\left(x_{1},x_{2},\ldots,x_{n},x\right)$ les points ordonnés $x_{1},\ldots,x_{n},x$ sont, ou .bien fixes ou bien on remplace $x_{1},x_{2},\ldots,x_{n}$ par des points aussi voisins qu’on veut ; on peut donc s’arranger toujours de manière que $\mathbb{V}\left(x_{1},x_{2},\ldots,x_{n},x\right)$ reste plus grand qu’un nombre positif.

Il en résulte que $A_{1}$ reste borné quand $x^{\prime}$ varie.
Par le même procédé nous obtenons

f\left(x^{\prime}\right)-f(x)\leq\left(x^{\prime}-x\right)\mathrm{B}_{1}

(76)

$\mathrm{B}_{4}$ étant borné lorsque $x^{\prime}$ varie. Pour celà il suffit de prendre par exemple $x_{n}$ à droite de $x$ . On procède de la même manière si $x^{\prime}$ est à gauche de $x$ .

Les inégalités (75), (76) prouvent la continuité.
Comme au No. précédent nous obtenons la propriété suivante :
Si une fonction bornée définie dans l’intervalle ( $a,b$ ) vérifie l’inégalité (72), elle est non-concave d’ordre $n$ (au sens du Chap. II).

Signalons encore la propriété suivante ( ⁴⁴ ) :
Si une fonction mesurable (au sens de M. Lebesgue) vérifie l’inégaLité (72), dans ( $a,b$ ) elle est continue en tout point intérieur.

Supposons le contraire et soit $x$ un point de discontinuité intérieur à ( $a,b$ ). Il résulte de ce qui précède que dans tout intervalle entourant $x$ il existe un point $\xi$ tel que

|f(\xi)|>\mathrm{A}

(77)

A étant un nombre positif aussi grand qu’on veut.
Soit a un nombre positif tel que l’intervalle ( $x-2\sigma,x+2\sigma$ ) soit complètement intérieur à ( $a,b$ ). Dans l’intervalle ( $x-\sigma,x+\sigma$ ) il existe un point $\xi$ tel que

|f(\xi)|>(n+1)\binom{n+1}{k}\mathrm{\penalty 10000\ A}

(78)

où $k$ est égal à $\frac{n}{2}$ ou $\frac{n+1}{2}$ suivant que $n$ est pair ou impair.

⁰⁰footnotetext: ( ⁴⁴ ) Pour

n=1

voir W. Sierpinski "Sur les fonctions convexes mesurables". Fundamenta Math. t. I (1920), p, 125.

Considérons l’un des intervalles de longueur $\sigma$ ayant $\xi$ comme extrémité. Nous avons
$\frac{1}{n+1}\left\{f(\xi-h)+\binom{n+1}{2}f(\xi+h)-\binom{n+1}{3}f(\xi+2h)+\cdots\right\}\geq f(\xi)\geq\geq\binom{n+1}{1}f(\xi+h)-\binom{n+1}{2}f(\xi+2h)+\binom{n+1}{3}f(\xi+3h)+\cdots$
en supposant par exemple $h>0$ et $\xi+(n+1)h\leq x+2\sigma$ .
On -voit alors qu’on a

|f(\xi+jh)|>\mathrm{A}

au moins pour un $j=-1,1,2,\ldots,n+1$ . En effet autrement il $y^{-}$ aurait certainement contradiction avec (78).

Il en résulte que les points $\xi$ , pour lesquels on a (77), forment un ensemble de mesure $\geq\sigma$ et alors la fonction ne peut être mesurable en vertu d’un théorème de M. Borel.

SECONDE PARTIE.

SUR LES FONCTIONS CONVEXES D’ORDRE SUPÉRIEUR DE DEUX VARIABLES RÉELLES.

CHAPITRE V.

SUR LES DIFFÉRENCES DIVJSÉES DES FONCTIONS DE DEUX variables réelles.

§ 1. - Théorie générale des différences divisées.

33.

Considérons une fonction $f(x,y)$ réelle et uniforme sur un en-semble plan borné $E$ dont la nature sera précisée plus loin.

On peut généraliser la notion de dufférence divicée pour le cas des fonctions de deux variables indépendantes d’une infinité de manières. Nous étudierons la généralisation qui paraît présenter le plus d’intérêt.

Soient $\mathrm{M}_{1},\mathrm{M}_{2},\ldots,\mathrm{M}_{k},k=(m+1)(n+1)$ points de l’ensemble E. Désignons par $x_{i},y_{i},i=1,2,\ldots,k$ les coordonnées du point $\mathrm{M}_{i}$ et posons

\mathrm{V}_{m,n}\left(\mathrm{M}_{1},\mathrm{M}_{2},\ldots,\mathrm{M}_{k}\right)=

(79)

$=\left|1x_{i}x_{i}^{2}\ldots x_{i}^{m}y_{i}y_{i}x_{i}y_{i}x_{i}^{2}\ldots y_{i}x_{i}^{m}\ldots y_{i}^{n}y_{i}^{n}x_{i}\quad y_{i}^{n}x_{i}^{2}\ldots y_{i}^{n}x_{i}^{m}\right|$
où, suivant une notation usitée, nous n’avons écrit que la ligne générale du déterminant.

Pour abréger, nous désignerons aussi par e l’ensemble des points. $\mathrm{M}_{i}$ et par $\mathrm{V}_{m,n}(e)$ le déterminant (79).

Désignons par $\mathrm{U}_{m,n}\left(\mathrm{M}_{1},\mathrm{M}_{2},\ldots\mathrm{M}_{k};f\right)$ ou $\mathrm{U}_{m,n}(e;f)$ le détérminant qu’on déduit de (79) en remplaçant les éléments de la dernière : colonne par

f\left(x_{1},y_{1}\right),f\left(x_{2},y_{2}\right),\ldots,f\left(x_{k},y_{k}\right)

Appelons courbe d’ordre ( $m,n$ ) une courbe algébrique représentée. par l’équation $\mathrm{F}(x,y)=0$ où $\mathrm{F}(x,y)$ est un polynome de degré $m$ ena $x$ et de degré $n$ en $y$ .

Considérons alors le quotient

\left[\mathrm{M}_{1},\mathrm{M}_{2},\ldots,\mathrm{M}_{k};f\right]_{m,n}=\frac{\mathrm{U}_{m,n}\left(\mathrm{M}_{1},\mathrm{M}_{2},\ldots,\mathrm{M}_{k};f\right)}{V_{m,n}\left(\mathrm{M}_{1},\mathrm{M}_{2},\ldots,\mathrm{M}^{k}\right)}

(80)

Remarqouns que la différence divisée d’ordre ( $m,n$ ) est symétrique par rapport aux points $\mathrm{M}_{i}$ .

Soit $\mathrm{E}^{*}$ l’ensemble dont les éléments sont les groupes de $k==(m+1)(n+1)$ points de E non situés sur une courbe d’ordre $(m,n)$ . A chaque élément de $\mathrm{E}^{*}$ correspond, pour la fonction $f,x,y$ ), une dififérence divisée d’ordre ( $m,n$ ).

Prenons un sous-ensemble $\mathrm{E}^{*}{}_{1}$ de $\mathrm{E}^{*}$ . L’ensemble de toutes les différences divisées d’ordre ( $m,n$ ) forme ce que nous appelerons une différence divisée d’ordre ( $m,n$ ) sur E de la fonction $f(x,y)$ .

La différence divisée d’ordre ( $m,n$ ) sur E est donc une fonction d’ensemble égale au quotient (80) en tout élément de $\mathrm{E}^{*}{}_{1}$ . Ainsi à tout sous-ensemble $\mathrm{E}_{1}^{*}$ correspond uue différence divisée d’ordre ( $m,n$ ) sur E.

Un sous-ensemble $\mathrm{E}^{*}$ : est toujours caractérisé par certaines propriétés restrictives que doivent vérifier les groupes de points donnant ales éléments de ce sous-ensemble.
34. Supposons que tout point de E appartienne à au moins un "élément de $\mathbb{E}_{1}^{*}$ . Nous dirons alors que la différence divisée d’ordre ( $m,n$ ) sur E , correspondant à $\mathrm{E}^{*}\mathrm{\penalty 10000\ s}$ , est complète.

Il est à peu près évident qu’une différence divisée sur. E non complète ne présente aucune utilité pour l’étude de la fonction sur E.

Soit $\mathrm{E}_{1}$ un groupe de $k=(m+1)(n+1)$ points de E , ce groupe étant un élément de $\mathrm{E}^{*}{}_{1}$ , $\mathrm{E}_{2}$ l’ensemble de tous les points de E tels que chacun d’eux donne, avec $k-1$ points de $\mathrm{E}_{1}$ , un élément de $\mathrm{E}_{4}^{*}\ldots,\mathrm{E}_{p}$ l’ ensemble de tous les points de E tels que chacun d’eux donne, avec ${}_{..}k-1$ points de $\mathrm{E}_{p-1}$ , un élément de $\mathrm{E}_{1}^{*},\ldots$ etc.

On a $\mathrm{E}_{1}<\mathrm{E}_{2}<\ldots<\mathrm{E}_{p}<\ldots$ et la somme $\Sigma\mathrm{E}_{p}$ est contenue dans E.

Supposons qu’on puisse trouver $\mathrm{E}_{\mathrm{f}}$ de manière qu’on ait $\Sigma\mathrm{E}_{p}=\mathrm{E}$ . - Nous dirons alors que la différence divisée sur E , correspondant à $\mathrm{E}^{*}\mathrm{t}$ , est close.

Il est clajr que la propriété d’être close entraine celle d’ètre complète.

Soient $\mathrm{E}_{1}^{*},\mathrm{E}_{2}^{*}$ deux sous-ensembles de $\mathrm{E}^{*}$ tels que $\mathrm{E}_{1}^{*}<\mathrm{E}^{*}{}_{2}$ Nous dirons alors que la différence divisée d’ordre ( $m,n$ ) sur E , correspondant au sous-ensemble $\mathrm{E}^{*}{}_{2}$ est plus élendue que celle correspondant au sous-ensemble $\mathrm{E}_{i}^{*}$ . On peut aussi dire que la différence divisée sur E , correspondant à $\mathrm{E}_{1}^{*}$ , est moins étendue que celle correspondant à $\mathrm{E}_{2}^{*}$ .

Si une différence divisée sur E est complète ou close, toute différence divisée sur E plus étendue sera a fortiori complète ou close.

Nous supposons bien entendu qu’il existe au moins une différence divisée d’ordre ( $m,n$ ) donc que l’ensemble E ne soit pas vide. Pourqu’il en soit ainsi il taut et il suffit que les points de E ne soient pas. tous sur une courbe d’ordre ( $m,n$ ).

Une différence divisée d’ordre ( $m,n$ ) sur E est nulle identiquement si les différences divisées sont nulles pour tout élément de l’ensemble. $\mathrm{E}_{1}^{*}$ correspondant.

Si une différence divisée close d’ordre ( $m,n$ If the function is zero on E, then the function reduces on E to the values of a polynomial of degree. $m$ in $x$ and degree $n$ in $y$ not containing a term in $x^{m}y^{n}$ .

Let us consider a sequence of sets $\mathrm{E}_{1},\mathrm{E}_{2},\ldots,\mathrm{E}_{p},\ldots$ expressing... the closure. We will demonstrate the property step by step. The property is true on the set $\mathrm{E}_{1}$ This is immediately apparent when noting that, among the $k=(m+1)(n+1)$ points of $\mathrm{E}_{1}$ , there is always $k-1$ through which passes a curved scule of order ( $m,n$ It now suffices to show that if the property is true on the set $\mathrm{E}_{p-1}$ it will also be true across the board $\mathrm{E}_{p}$ This results from the fact that the points of $\mathrm{E}_{p}$ which are not found in $\mathrm{E}_{p-1}$ are obtained from there in the following way: We take in $\mathrm{E}_{p-1}k-1$ points through which a single curve of order ( $m,n$ ) ; either ( $C$ ) this curve. We take the points that are not on (C) and that with the $k-1$ points. The chosen points give an element of the set $\mathrm{E}^{*}{}_{1}$ corresponding to the difference divided over E considered. By taking all possible curves (C), we obtain all the points of $\mathbf{E}_{p}$ 35.
Let us establish yet another property of certain divided differences on E. For a divided difference on E to be usable for the study of functions, it is necessary that, at least for simple functions, it leads to simple differential properties for the function. $f(x,y)$ .

Either $M^{\prime}$ a point on the derivative $E^{\prime}$ of $E$ and an element of $E^{*}{}_{1}$ formed
by the points $\mathrm{M}_{i},i=1,2,\ldots,k,k=(m+1)(n+1)$ Let's call the distance from the point $\mathrm{M}^{\prime}$ to the element e the greatest of the distances from the point $\mathrm{M}^{\prime}$ to the points $M_{i}$ .

Let's agree that the sequence of elements of $\mathrm{E}^{*}{}_{1},e^{\prime},e^{\prime\prime},\ldots,e^{(p)},\ldots$ / tends towards the point $\mathrm{M}^{\prime}$ of $\mathrm{E}^{\prime}$ if the distance from the point $\mathrm{M}^{\prime}$ to the element $e^{(p)}$ tends towards zero when $p\rightarrow\infty$ .

We will say that a difference divided by E is regular if, for whatever reason, a sequence of elements $e^{\prime},e^{\prime\prime},\ldots,e^{\prime p},\ldots$ of E tends towards the point $\mathrm{M}^{\prime}$ of $\mathrm{E}^{\prime}$ the difference divided $\left[e^{(p)};x^{\alpha}y^{\beta}\right]_{m,n}$ tends towards a finite and well-defined limit, and this:
10. For any point M' which is the limit of at least one sequence of elements of $E_{1}^{*}$
$2^{0}$ For a couple of numbers $\alpha,\beta$ non-negative integers ( ⁴⁵ ).
The regularity conditions are not all independent. First, for $\alpha\leq m,\beta\leq n,\alpha+\beta<m+n$ the limit in question is always equal to zero and for $\alpha=m,\beta=n$ It is obviously equal to 1. These properties belong to all differences divided over E.

Let us designate by $\mathrm{L}_{m,n}^{(\alpha,\beta)}\left(x^{\prime},y^{\prime}\right)$ the limit of the difference divided $\left[e;x^{a}y^{\beta}\right]_{m,n}$ to the point $M^{\prime}\left(x^{\prime},y^{\prime}\right)$ , in the case of regularity. Therefore, we have

	$\displaystyle\mathrm{L}_{m,n}^{\left(a_{1}\beta\right)}\left(x^{\prime},y^{\prime}\right)$	$\displaystyle=0$		$\displaystyle\alpha\leq m,\quad\beta\leq n,\alpha+\beta<m+n$
		$\displaystyle=1$		$\displaystyle\alpha=m,\quad\beta=n.$

We can write

\left[e;x^{\alpha}y^{\beta}\right]_{m,n}=\sum_{\imath=1}^{k}A_{i}c_{\imath}^{\alpha}y_{\imath}^{\beta}\quad(k=(m+1)(n+1))

Your $A_{i}$ being independent of $\alpha$ And $\beta$ .
Let us designate by $\mathrm{X}_{i}$ the fundamental symmetric functions of the abscissas of the points forming $e$ and by $\mathrm{Y}_{i},i=i,2,\ldots,k$ the functions

⁰⁰footnotetext: ⁽⁴⁵⁾ Les considérations précédentes s’appliquent aux fonctions d’une variable ^∗ La différence divisée générale, envisagée dans la première partie, est close. Si la fonction est définie dans un intervalle, la différence divisée prise sur des points équidistants est complète mais n’est pas close. La question de régularité ne se pose pas. Toute différence divisée est régulière. C’est précisement à cause de cette propriété que les questions exposées dans la première cpartie presentaient une aussi grande simplicité.

fundamental symmetrics of the ordinates of these points. We have

	$\displaystyle{\left[e;\mathfrak{x}^{a}y^{\beta}\right]_{m,n}}$	$\displaystyle=\sum_{\imath=1}^{k}(-1)^{i-1}X_{i}\left[e;\mathfrak{x}^{\alpha-i}y^{\beta}\right]_{m,n}$		$\displaystyle\alpha\geq k$
		$\displaystyle=\sum_{i=1}^{k}(-1)^{i-1}Y_{i}\left[e;\mathfrak{x}^{\alpha}y^{\beta-i}\right]_{m,n}$		$\displaystyle\beta\geq k$

This shows us that:
There are only a finite number of independent regularity conditions.

In particular, it is sufficient that the regularity condition be met for $\alpha\leq k-1,\beta\leq k-1$ and then we have

	$\displaystyle L_{m,n}^{(\alpha,\beta)}\left(x^{\prime},y^{\prime}\right)$	$\displaystyle=\sum_{i}^{k}(-1)^{i-1}\binom{k}{i}\mathfrak{X}^{\prime i}L_{m,n}^{(\alpha-i,\beta)}\left(\mathscr{X}^{\prime},y^{\prime}\right)\quad\alpha\geq k$
		$\displaystyle=\sum_{i=1}^{k}(-1)^{i-1}\binom{k}{i}y^{\prime i}L_{m,n}^{(\alpha,\beta-i)}\left(\mathscr{A}^{\prime},y^{\prime}\right)\quad\beta\geq k$

We have thus found $(m-1)(n+1)(m+n+mn)$ Conditions of regularity. These conditions are not all independent yet and can be reduced in various ways. We will not dwell on this point here.

Regularity does not lead to closure and conversely closure does not lead to regularity ( ⁴⁶ ).

If a difference divided over E is regular, any less extensive difference divided over E is also regular and exhibits the same regularity (the limits are the same).

§. 2. - On a particular divided difference.

36.

If the subset $\mathrm{E}^{*}{}_{1}$ coincides with E, we obtain the most extensive divided difference over E. Since E is not empty, it can easily be shown that this divided difference over E is closed. However, this divided difference over E does not appear to be of great interest for the study of the function, as it is not generally smooth. Let

⁰⁰footnotetext: (46) C’est précisement sur ce point que le cas de deux variables diffère essentiellement de celui d’une variable.

for example the four points

\begin{gathered}\mathrm{M}_{1}(1,1),\quad\mathrm{M}_{2}\left(1-a,1-a^{2}(1-\theta)\right),\quad\mathrm{M}_{3}\left(1+a,1-a,1-a^{2}(1-\theta)\right)\\ \mathrm{M}_{4}\left(1+a^{2},1+\theta a^{2}\right)\end{gathered}

and the function $f(x,y)=x^{2}$ ; We have

\left[M_{1},M_{2},M_{3},M_{4};f\right]_{1,1}=(1-\theta)a^{2}+\theta

If $a\rightarrow 0$ the four points tend towards the point $(1,1)$ and the divided difference can tend towards any limit.
37. We will assume, for simplicity, that E is a closed rectangle ( ⁴⁷ )

\mathrm{E}\binom{a\leq x\leq b}{c\leq y\leq a}

(81)

Let's call with Mr. Marchaud $\left({}^{48}\right)$ , order network ( $m,n$ ) the figure formed by $m$ parallel to the axis $0y$ And $n$ parallel to the axis $0x$ Of course, we only consider the points of the network that belong to rectangle E.

We call the partial divided difference of order ( $m,n$ ) the difference divided over E corresponding to the subset $\mathrm{E}^{*}{}_{1}$ whose elements are the nodes (or points of intersection) of all the networks. of order ( $m+1,n+1$ ).

The partial divided difference of order ( $m,n$ The system is complete but not closed. It is also regular, and its regularity is expressed by the equalities.

\mathrm{L}_{m,n}^{(\alpha,\beta)}\left(x^{\prime},y^{\prime}\right)=\binom{\alpha}{m}\binom{\beta}{n}x^{m-\alpha}y^{\prime m-\beta}.

The term "partial divided difference" can be explained as follows: Let's take the same divided difference of the function with respect to $x$

\mathrm{F}(y)=\left[x_{1},x_{2},\ldots,x_{m+1};f\right]

and the nth difference divided by $\mathrm{F}(y)$

\left[\begin{array}[]{l}x_{1},x_{2},\ldots,x_{m+1}\\ y_{1},y_{2},\ldots,y_{n+1}\end{array};f\right]=\left[y_{1},y_{2},\ldots,y_{n+1};F\right].

By changing the order of the two variables $x,y$ we also define the quantity

\left[\begin{array}[]{l}y_{1},y_{2},\ldots,y_{n+1}\\ x_{1},x_{2},\ldots,x_{m+1}\end{array};f\right].

⁽⁴⁷⁾ It is clear that a more complicated set could be taken. (48) Thesis loc. cit. (7).

We can easily see that we have identical

\left[\begin{array}[]{l}y_{1},y_{2},\ldots,y_{n+1}\\ x_{1},x_{2},\ldots,x_{m+1}\end{array};f\right]=\left[\begin{array}[]{l}x_{1},x_{2},\ldots,x_{m+1}\\ y_{1},y_{2},\ldots,y_{n+1}\end{array};f\right].

Expression (82) is precisely the divided difference of order $(m,n)$ of the function $f(x,y)$ on the points $\left(x_{i},y_{i}\right),i=1,2,\ldots,m+1j=1,2,\ldots,n+1$ .

Let us also call, with Mr. Marchaud (49), a pseudo-polynomial of order ( $m,n$ ) any function of the form

\sum_{i=0}^{m}x^{i}\mathrm{\penalty 10000\ A}_{i}(y)+\sum_{j=0}^{n}y^{\prime}\mathrm{B}_{j}(x)\quad\begin{aligned} &\mathrm{A}_{l}(y)\text{ fonctions arbitraires de }y\\ &\mathrm{\penalty 10000\ B}_{j}(x)\text{ fonctions arbitraires de }x\end{aligned}.

A pseudo-polynomial of order $(m,n)$ is completely determined by its values on an ordered network ( $m+1,n+1$ ).

This property is analogous to the uniqueness property of L polynomials.

Let us designate by

\mathrm{P}\left(\begin{array}[]{l}x_{1},x_{2},\ldots,x_{m+1}\\ y_{1},y_{2},\ldots,y_{n+1}\end{array};f\binom{x}{y}\right.

the pseudo-polynomial of order $(m,n)$ coinciding with the function $f(x,y)$ on the network

\begin{array}[]{ll}x=x_{i},&i=1,2,\ldots,m+1\\ y=y_{j},&j=1,2,\ldots,n+1\end{array}.

It is easy to see that

	$\displaystyle f(x,y)-\mathrm{P}\left(\begin{array}[]{l}x_{1},x_{2},\ldots,x_{m+1}\\ y_{1},y_{2},\ldots,y_{n+1}\end{array};f\left\lvert\,\begin{array}[]{l}x\\ y\end{array}\right.\right)=$		(83)
	$\displaystyle\frac{\mathrm{V}\left(x_{1},x_{2},\ldots,x_{m+1}\right)\cdot\mathrm{V}\left(y_{1},y_{2},\ldots,y_{n+1}\right)}{\mathrm{V}\left(x_{1},x_{2},\ldots,x_{m+1},x\right)\cdot\mathrm{V}\left(y_{1},y_{2},\ldots,y_{n+1},y\right)}\left[\begin{array}[]{l}x_{1},x_{2},\ldots,x\\ y_{1},y_{2},\ldots,y\end{array}\right]\cdot$

38.

It can easily be seen that the general solution of the equation

\left[\begin{array}[]{l}x_{1},x_{2},\ldots,x_{m+1}\\ y_{1},y_{2},\ldots,y_{n+1}\end{array};f\right]=0.\quad\text{ sur }\mathrm{E}

is a pseudo-polynomial of order ( $m-1,n-1$ ).

		$\displaystyle\text{ (49) Nous appelons pseudo-polynome d'ordre }(m,n)\text{ ce que M. Marchaud }$
		$\displaystyle\text{ appelle d'ordre }(m+1,n+1)\text{. Nous introduisons ce changement en vue d'ob- }$
		to maintain greater symmetry in the names.

Mathemat ca VIII. $\square$

Partial divided differences of order $\left(m_{1},n_{1}\right)<(m,n)\left[m_{1}\leq m,n_{1}\leq n,m_{1}+n_{1}<m+n\right]$ of a pseudo-polynomial of order ( $m-1,n-1$ ) are generally unbounded. The pseudo-polynomial itself is generally unbounded. We have the following property:

For a pseudo-polynomial of order ( $m-1,n-1$ ) has all its differences divided by partial order $\leq(m,n)$ bounded it is necessary and sufficient that its divided partial differences of order ( $m,0$ ), ( $0,n$ ) are bounded.

Suppose the function $f(x,y)$ zero on the network

x=x_{i}^{\prime},\quad i=1,2,\ldots,m;\quad y=y_{j}^{\prime},\quad j=1,\ldots,n

Let's apply formula (6) to partial divided differences

		$\displaystyle{\left[\begin{array}[]{l}x_{1},x_{2},\ldots,x_{m}\\ y_{1},y_{2},\ldots,y_{n+1}\end{array};f\right]=\left[\begin{array}[]{l}x_{1},x_{2},\ldots,x_{m}\\ y_{1},y_{2},\ldots,y_{n+1}\end{array};f\right]-\left[\begin{array}[]{l}x_{1}^{\prime},x_{2}^{\prime},\ldots,x_{m}^{\prime}\\ y_{1},y_{2},\ldots,y_{n+1}\end{array};f\right]}$
		$\displaystyle\left.\left[\begin{array}[]{l}x_{1},x_{2},\ldots,x_{m+1}\\ y_{1},y_{2},\ldots,y_{n}\end{array};f\right]=\left[\begin{array}[]{l}x_{1},x_{2},\ldots,x_{n+1}\\ y_{1},y_{2},\ldots,y_{n}\end{array};f\right]-\left[\begin{array}[]{l}x_{1},x_{2},\ldots,x_{m+1}\\ y_{1}^{\prime},y_{2}^{\prime},\ldots,y_{n}^{\prime}\end{array}\right]f\right]$

we then see that if lx partial divided difference of order ( $m,n$ ) is bounded the partial divided differences of order ( $m-1,n$ ), ( $m,n-1$ ) are also limited.

It follows that all partial divided differences of order $\leq(m,n)$ are bounded. We can therefore state the following property:

So that the partial divided differences of order $\leq(m,n)$ of a function $f(x,y)$ to be bounded it is necessary and sufficient that:
10. The partial divided difference of order ( $m,n$ ) is bounded.
^20. Partial divided differences of order $(m,0),(0,n)$ are bounded on a network of order ( $m+1,n+1$ ).

We also see that:
Any function whose partial divided difference of order ( $m,n$ ) is bounded is the sum of a function having all its differences divided by partial order $\leq(m,n)$ bounded and of a pseudo-polynomial of order $(m-1,n-1)$ .

The fact that the partial divided difference of order $(0,0)$ is bounded means that the function is bounded.

It can also be noted that

f\left(x,y-f\left(x^{\prime},y^{\prime}\right)=\left(y-y^{\prime}\right)\left[\begin{array}[]{l}x\\ y,y^{\prime}\end{array};f\right]+\left(x-x^{\prime}\right)\left[\begin{array}[]{l}x,x^{\prime}\\ y\end{array}\right]f\right]

therefore, if the partial divided differences of order $(1,0)$ And $(0,1)$ are bounded the function is continuous.

Let us recall once again a theorem of MP Montel (50) under a slightly modified farm.

If the partial divided differences of order ( $m-m^{\prime},n$ ), ( $m,n-n^{\prime}$ ) of a ? function are bounded, the partial divided difference of order ( $m^{\prime\prime},n^{\prime\prime}$ ) is bounded provided that

\begin{array}[]{ccc}m^{\prime\prime}\geq m-m^{\prime}&m\geq m^{\prime}>0&\frac{m-m^{\prime\prime}}{m^{\prime}}+\frac{n-n^{\prime\prime}}{n^{\prime}}>1.\end{array}

We first show that the property is true for a pseudo-polynomial of order ( $m-1,n-1$ ) (the conditions $m\geq m^{\prime\prime}\geq m-m^{\prime}$ , $n\geq n^{\prime\prime}\geq\imath-n^{\prime}$ are then sufficient). We then demonstrate the property for a function that vanishes on a lattice of order ( $m,n$ ) (the first two conditions can then be replaced by $m^{\prime\prime}\leq m,n^{\prime\prime}\leq n$ The demonstration can be done using the functions $\omega_{n,n}(\delta,\lambda)$ introduced by Mr. Marchaud in his Thesis ( ⁵¹ ).

S. 3. - Study of another particular divided difference.

39.

To simplify, we will again assume that E is a rectangle (81).

Either $\mathrm{E}_{1}^{*}$ the subset of $\mathrm{E}^{*}$ whose elements are all the groups $e$ of $m+1$ points $\mathrm{M}_{i}\left(x_{i},y_{i}\right)$ meeting the following conditions:
$1^{0}$ The sequel $x_{1},x_{2},\ldots,x_{m+1}$ is ordered.
$2^{0}$ . We have

\left|y_{i+1}-y_{i}\right|\leq\phi\left(x_{i+1}-x_{i}\right),\quad\dot{\imath}=1,2,\ldots,m

Or $\phi(\theta)$ is a positive and non-decreasing function for $\theta>0$ We
will say that the difference divided by order ( $m,0$ ) on E corresponding to $\mathrm{E}_{1}^{*}$ is a normal divided difference of order ( $m,0$ The function $\phi(\theta)$ is its characteristic function.

A normal difference of order ( $m,0$ ) is closed; it is aregular if $\phi(\theta)$ is quite small and tends towards zero quite rapidly.

We will say that a difference divided over E is bounded at the point $\mathrm{M}(x,y)$ of E if there exists a circle centered at M where this divided difference is bounded. We also say that the divided difference over E is not bounded at point M if it is not bounded in any circle centered at M.

Let's demonstrate the following property:
If a tonction $f(x,y)$ has a normal divided difference of order

⁰⁰footnotetext: (50) Voir loc. cit. (35).
(51) Voir le troisième Chapitre de sa Thèse.

( $m,0$ ) bounded in E, it also has a normal divisible difference of order: ( $m-1,0$ ) bounded at every point of E.

—

We can demonstrate this using formula (6) from the first part, which is applicable here in the form

		$\displaystyle{\left[\mathrm{M}_{1},\mathrm{M}_{2},\ldots,\mathrm{M}_{m};f\right]_{m-1,0}-\left[\mathrm{M}_{1}^{\prime},\mathrm{M}_{2}^{\prime},\ldots,\mathrm{M}_{m}^{\prime};f\right]_{m-1,0}}$		(84)
	$\displaystyle=$	$\displaystyle\sum_{i=1}^{m}\left(x_{i}-x_{i}^{\prime}\right)\left[\mathrm{M}_{i},\mathrm{M}_{i+1},\ldots,\mathrm{M}_{m},\mathrm{M}_{1}^{\prime},\mathrm{M}_{2}^{\prime},\ldots,\mathrm{M}_{i}^{\prime};f\right]_{m,0}$

Or ( $x_{i},y_{i}$ ), ( $x_{i}^{\prime},y_{i}^{\prime}$ ) are the coordinates of the points $\mathrm{M}_{i},\mathrm{M}_{i}^{\prime}i=1_{i},2_{ij}\ldots,m$ .

Either $M(x,y)$ a point of $E$ and consider a circle with center $M$ and radius $p$ Let's take the fixed points $\mathrm{M}^{\prime};{}_{\mathrm{a}}$ i outside the circle and on the same side of the vertical line passing through M. Suppose we take them to the right of this vertical line so that the following $x^{*}{}_{1},x^{\prime}{}_{2},\ldots,x^{\prime}{}_{m}$ Let's consider the points. $\mathrm{M}_{i}$ within the circle as follows: $x_{1},x_{2},\ldots,x_{m}$ be ordered. The sequences $x_{i},x_{i+1},\ldots,x_{m},x_{1}^{\prime},x_{2}^{\prime},\ldots,x_{i}^{\prime}$ will then be ordered.

We can then see that we can always fix the points $\mathrm{M}^{\prime}{}_{i}$ and take the ray $\rho$ small enough so that whatever the points Misverifying the condition $2^{0}$ , the groups of points $\mathrm{M}_{i},\mathrm{M}_{i+4},\ldots,\mathrm{M}_{na}\mathrm{M}_{1}^{\prime},\mathrm{M}_{2}^{\prime},\ldots,\mathrm{M}_{i}^{\prime},i=1,2,\ldots,m$ also verify the condition $2^{0}$ Formula (84) then demonstrates the property.

We will specify the results obtained:
Let us denote by $\mathrm{C}(\mathrm{M};\mathrm{p})$ the circle with center M and radius $p$ We
can say that there is a $\rho>0$ such that the function has a normal divided difference of order $(m-1,0)$ barnée in $\mathrm{C}\left(\mathrm{M};\mathrm{Q}_{q}\right)_{2}$ regardless of point M.

For all points M located to the left of the vertical line with abscissa $\frac{a+b}{2}$ we take the points $\mathrm{M}_{i}^{\prime}$ such that one has $x_{i}^{\prime}<b-\lambda,i=1,2;\ldots,m$ , $\lambda$ being a sufficiently small fixed positive number. The property follows for all these points, taking into account that the characteristic function $\phi(\theta)$ of the normal divided difference of order ( $m,0$ The given value is decreasing. The same applies to points M that are to the right of the vertical line with abscissa $\frac{a+b}{2}$ The stated property results from this.

I say again that: the difference divides normal diorder ( $m-1,0$ ) is uniformly bounded in circles $\mathrm{C}\left(\mathrm{M};\rho_{1}\right),\rho>\rho_{1}>0$ .

This statement means that there exists a positive number $\mathrm{A}_{2}$ , such as damæ:
the circle $C\left(M;p_{1}\right)$ the divided differences of the normal divided difference considered, do not exceed in modulus the number A, whatever M.

Indeed, otherwise it is easy to find that there is a point $M_{0}$ such as in the circle $C\left(M_{0};\rho\right)$ the normal divided difference of order ( $m-1,0$ ) is not bounded; which is impossible.
40. Suppose that the function $f(x,y)$ has a normal divided difference of order ( $m,0$ ) which is bounded within the circle $\mathrm{C}(\mathrm{M};\rho),\rho>0$ -whatever M in E.

We will show that in this case the function has a (different) normal divided difference of order ( $m,0$ ) bounded in rectangle E.

Either $\phi(\theta)$ the characteristic function of the given normal divided difference. Consider the normal divided difference of order ( $m,0$ ) whose characteristic function $\phi_{1}(\theta)$ is defined as follows:

\begin{array}[]{lll}\phi_{1}(\theta)=\phi\left(\rho_{1}\right)&\text{ pour }0\leq\theta\leq\frac{\rho_{1}}{2m}&\rho>\rho_{1}>0\\ \phi_{1}(\theta)=\phi\left(\rho_{1}\right)&\text{ pour }\theta>\frac{\rho_{1}}{2m}.&\end{array}

We will show that there is always a choice $\rho_{1}$ so that "this normal divided difference satisfies the property.

Let $\mathrm{M}_{i}\left(x_{i},y_{i}\right),i=1,2,\ldots,m+1,m+1$ points in E satisfying inequalities:

		$\displaystyle x_{1}<x_{2}<,\ldots,<x_{m+1}$
		$\displaystyle\left\|y_{i+1}-y_{i}\right\|\leq\phi_{1}\left(x_{i+1}-x_{i}\right),\quad i=2,\ldots,m.$

On each pair of "consecutive" points $\mathrm{M}_{i},\mathrm{M}_{i+1}$ we perform the following operation: If $x_{i+1}-x_{i}\leq\frac{p_{1}}{m}$ we leave the points unchanged, if $\frac{\rho_{1}}{m}<x_{i+1}-x_{i}\leq\frac{2\rho_{1}}{m}$ we share the segment $M_{i}M_{i+1}$ in two equal parts by an additional point; in general, if $\frac{(j-1)\rho_{1}}{c}<x_{i+1}-x_{i}\leq\frac{j\rho_{1}}{m}$ we share the segment $M_{i}M_{i+1}$ in $j$ - equal parts by $j-1$ extra points.

Let's tidy up the points $\mathrm{M}_{i}$ and all the additional points introduced in the order of growth of their x-coordinates. We then see that we can choose $\rho_{1},\rho_{2}\left(\rho>\rho_{2}>\rho_{1}>0\right)$ , so that whatever the points $\mathrm{M}_{i}$ any group of $m+1$ consecutive points of the obtained sequence satisfy the following properties:
10. The group of $m+1$ points is an element of the subset $\mathrm{E}^{*}{}_{1}$ correspond to the given normal divided difference (whose characteristic function is $\phi(\theta)$ ).
$2^{0}$ . THE $m+1$ points of the group are in a circle of radius $p_{2}$ having as its center a point of $\mathbf{E}$ .

The stated property then follows, possibly applying formula (10) from the first part and taking into account the fact that the given normal divided difference is uniformly bounded in circles. $C\left(M;p_{2}\right)$ .

Taking into account the results of the previous No., we can state the property:

If a function $f(x,y)$ has a normal divided difference of order ( $m,0$ ) bounded in E, it also has a normal divided difference of order. ( $m-1,0$ ) bounded in E.

We also see that the function has a normal divided difference of order ( $r,0$ ) bounded in E for $r<m$ .

Note. Consider the normal divided difference of order ( $m,0$ ) whose characteristic function is $\lambda.\theta,\lambda$ being a positive number _∘ For this normal divided difference to be bounded in E, it is necessary and sufficient that it be bounded at every point of E. Indeed, it can be shown, using formula (10), that if it is not bounded in E, there exists a point where it is unbounded.
41. If the function $f(x,y)$ has a normal divided difference of order $(1,0)$ bounded, it is continuous at every point of $\mathbb{E}$ .

Either $M(x,y)$ a point of $E$ And $M_{p}\left(x_{p},y_{p}\right)$ a sequence of points in E tending towards M. We see that for every number $\varepsilon>0$ we can match a number $N$ and an abscissa $x^{\prime}{}_{p}$ such that one has

\left|f(x,y)-f\left(x_{p}^{\prime},y_{p}\right)\right|<\frac{\varepsilon}{2},\left|f\left(x_{p}^{\prime},y_{p}\right)-f\left(x_{p},y_{p}\right)\right|<\frac{\varepsilon}{2}

For $p>\mathrm{N}$ ; SO

\left|f(x,y)-f\left(x_{p},y_{p}\right)\right|<\varepsilon,\quad p>N

If the function has a normal divided difference of order ( $m,0$ ) bounded, it has at every point of E a partial derivative of order $m-1$ , $\frac{\partial^{m-1}f}{\partial x^{m-1}}$ We intend to show that:

If the function $f(x,y)$ has a nonmale divided difference of order $(m,0),m>1$ bounded, the partial derivative $f_{x}^{\prime}$ exists at every point and has a normal divided difference of order ( $m-1,0$ ) limited.

To demonstrate the property, it suffices to take the points $M_{i},M_{i}^{r}{}_{i}^{r}$
so that $y_{1}=y_{i}^{\prime},\quad i=1,2,\ldots,m$ And $x_{1}<x_{1}^{\prime}<x_{2}<x_{2}^{\prime}<\ldots$ We can then apply the reasoning used for functions of one variable in Chapter III. By letting tend $x_{i}^{\prime}$ towards $x_{i},i=1,2,\ldots,m$ We can see that $f^{\prime}{}_{x}$ has a normal divided difference of order ( $m-1,0$ ) bounded corresponding to the same characteristic function $\phi(\theta)$ .

From the previous property and from what was said in Nos. 39 and 40, we deduce that:

If the function $f(x,y)$ has a normal divided difference of order $(m,0)$ bounded, it has partial derivatives $f^{\prime}x,f^{\prime\prime}x^{2},\ldots,f_{x^{m-1}}^{(m-1)}$ continuous at every point of E.

The derivative $f_{x^{i}}^{(i)}$ has a normal divided difference of order ( $m-i,0$ ) bounded and in particular $f_{xm-1}^{(m-1)}$ has a normal divided difference of order $(1,0)$ narrow-minded.

Suppose that the function $f(x,y)$ has a normal divided difference of order ( $m,0$ ) and that this normal divided difference is such that the divided difference has a finite and well-determined limit when the points on which it is taken tend in any way towards a limit point.

It follows that the normal divided difference considered is bounded and that the partiolle derivative $\frac{\partial^{m}f}{\partial x^{m}}$ exists and is continuous. Conversely, if $\frac{\partial^{m}f}{\partial x^{m}}$ exists and is continuous the normal divided difference function of order ( $m,0$ ) bounded. It should be noted that the characteristic function $\phi(\theta)$ that of this normal divided difference generally depends on the function considered.

We can also demonstrate the following property:
For the tonction $f(x,y)$ has a partial derivative $\frac{\partial^{m}f}{\partial x^{m}}$ continuous at every point of E, it is sufficient that there exists a normal divided difference of order ( $m,0$ ) such that no matter how the points on which a divided difference is taken tend towards a limit point, the divided difference tends towards a finite and well-determined limit.

Let's add that we can make the same construction for the order $(0,n)$ and thus introduce normal divided differences of order ( $0,n$ ) which will enjoy the same properties with respect to the variable $y$ 42.
Let's define the subset $\mathrm{E}_{1}^{*}$ so that its elements $e$ all groups of $k=(m+1)(n+1)$ points $M_{i,j}\left(x_{i,j},y_{i,j}\right)$ , $i=1,2,\ldots,m+1,j=1,2,\ldots,n+1$ meeting the following conditions:

1'. The sequels

\begin{array}[]{lr}x_{1,j},x_{2,j},\ldots,x_{m+1,j}&j=1,2,\ldots,n+1\\ y_{i,1},y_{i,2},\ldots,y_{i,n+1}&i=1,2,\ldots,m+1\end{array}

are ordered.
$2^{0}$ We have inequalities

\begin{array}[]{lll}\left|y_{i+1,j}-y_{i,j}\right|\leq\phi\left(x_{i+1,j}-x_{i,j}\right),&i=1,2,\ldots,n&j=1,2,\ldots,m+1\\ \left|x_{i,j+1}-x_{i,j}\right|\leq\phi\left(y_{i,j+1}-y_{i,j}\right),&i=1,2,\ldots,m&j=1,2,\ldots,n+1\end{array}

Or $\phi(\theta)$ is a positive and non-decreasing function for $\theta>0$
We can determine the function $\phi(\theta)$ so that we also have $\mathrm{V}_{m,n}(e)\neq 0$ , whatever the points $\mathrm{M}_{i,j}$ We will assume that this is always the case. We see that if the function $\phi(\theta)$ checks this property; any smaller function will check the same property.

We call the divided difference of order (m, n) the divided difference on E corresponding to such a subset $\mathrm{E}^{*}{}_{1}$ . The function $\phi(\theta)$ is its characteristic function.

A normal divided difference of order ( $m,n$ ) is more extensive than the partial divided difference of the same order. Therefore, any normal divided difference is complete.

Let us further demonstrate that every normal divided difference is closed.
Indeed, let $\mathrm{P}_{i,}\left(x_{i},y_{j}\right)i=1,2,\ldots,m+1j=1,2,\ldots,n+1$ the nodes of an ordered network ( $m+1,n+1$ ). Let us designate by $e$ all of these $k=(m+1)(n+1)$ Points attached to each PCint $P_{i,j}$ a circle having this point as its center and a radius the positive number $p$ Let us consider the sets $e^{\prime}$ of $k$ points $\mathrm{P}^{\prime}{}_{i,}$ , such that each is in or on the circumference of the circle corresponding to the point $\mathrm{P}_{i,j}$ same indices $i$ And $j$ There is a $\rho_{e}$ depending on the set $e$ such as $:1^{0}\mathrm{si}\rho<\rho$ we have $\mathrm{V}_{m,n}\left(e^{\prime}\right)\neq 0$ regardless of $e^{\prime}$ ; $2^{0}$ if $\rho\geq\rho_{e}$ there is a $e^{\prime}$ whose determinant $\mathrm{V}_{m,n}$ is zero. $\rho_{e}$ is independent of a translation of the point group $\mathrm{P}_{i,/}$ .

Let us now consider the subset $\mathrm{E}^{*}$ , the elements of which are $e^{\prime}$ obtained by taking for $\rho$ a value $<\rho_{e}$ independent of translation. It can easily be shown that the divided difference on E corresponding to such a subset is closed, by expressing the closure condition in a suitable way. Now, given a normal divided difference, there always exists a divided difference on E of the latter form that is less extensive, which proves the property.

We can state the following property:

If a normal divided difference of order ( $m,n$ ) of a function s'zero identically, this function reduces on E to a polynomial of degree $m$ in $x$ and degree $n$ in $y$ not containing a term in $x^{m}y^{n}$ .

Without going into details, let's just say that it can be shown that if the characteristic function tends sufficiently rapidly towards zero with $\theta$ , the normal divided difference is also regular. This regularity is obviously always that of the partial divided difference of the same order.
43. Consider a normal divided difference of order ( $m,n$ ). :Either $\mathrm{E}^{*}{}_{1}$ the subset corresponding to this normal divided difference and $\phi(\theta)$ its characteristic function. We assumed that points of an element of E* ₁ are all distinct. Suppose that a sequence of elements of $\mathrm{E}^{*}{}_{1}$ tends towards a borderline group $e$ of $(n+1)(n+1)$ points and are $\mathrm{M}_{i,j}\left(x_{i,j},y_{i,j}\right)i=1,2,\ldots,m+1,j=1,2,\ldots,n+1$ the points of $e$ such as

\begin{array}[]{ll}x_{1,j}\leq x_{2,j}\leq\ldots\leq x_{m+1,j}&j=1,2,\ldots,n+1\\ y_{i,1}\leq y_{i,2}\leq\ldots\leq y_{i,n+1}&i=1,2,\ldots,m+1\end{array}

The points $e$ are not necessarily all distinct. It can easily be seen that if the characteristic function $\phi(\theta)$ tends quite rapidly towards zero with $\theta$ , a multiple point of $e$ is always formed by a group of $(p+1)(q+1)$ points combined such as $\mathrm{M}_{r+i,s+\text{ J }}i=0,1,2,\ldots,p,j=0,1,2,\ldots,q$ .

We will try to make sense of the difference divided for such a group $e$ , containing coincident points. Let's always assume that the points $\mathrm{M}_{r+i,s+j}0\leq i\leq p,0\leq j\leq q$ come together at the point $\mathrm{M}(x,y)$ Let's assume they are distinct at first. We substitute into $\mathrm{U}_{m,n}(e;f)$ the lines corresponding to the points $\mathrm{M}_{r+i,s+j}$ by $(p+1)(q+1)$ other lines that can be deduced from the line

1xx^{2}\ldots x^{m}yyxyx^{2}\ldots yx^{m}\ldots y^{n}y^{n}x\ldots y^{n}x^{m}\quad f(x,y)

by the following procedure: The line corresponding to the point is replaced $\mathrm{M}_{r+i,s+/}$ by (80) in which each term has been substituted for its divided difference of order ( $i,j$ ) taken on the points $\mathrm{M}_{r+\alpha,s+\beta}\alpha=0,1,2,\ldots,i,\quad\beta=0,1,2,\ldots,j$ We perform this operation because $i=0,1,2,\ldots,p,j=0,1,2,\ldots,q$ [we are asking of course $\left[\mathrm{M}_{r,s};f\right]_{0,0}==f\left(x_{r,s},y_{r,s}\right)$ Let's now tighten the points $\mathrm{M}_{r-i,s+j}$ towards $\mathrm{M}\left({}^{52}\right)$ It is then easy to see that if the characteristic function tends sufficiently
( ⁵² ) We assume of course that these points constantly belong to elements of $E^{*}{}_{1}$ .
quickly towards zero with $\theta$ , the line corresponding to the point $\mathrm{M}_{r+i,s+j}$ tends towards what we deduce from (85), if we apply the operation to its terms: $\frac{\partial^{i+j}}{\partial x^{i}\partial y^{j}}$ provided, of course, that the derivative $\frac{\partial^{i+i}f}{\partial x^{i}\partial y^{j}}$ exists and is continuous. We can choose $\phi(\theta)$ so that this is true for $i=0,1,2,\ldots,p,j=0,1,2,\ldots,q$ .

Finally, we can choose $\phi(\theta)$ so that the previous process applies to all multiple points of $e$ provided that all the introduced derivatives exist and are continuous. The determinant $\mathrm{V}_{m,n}(e)$ ; will be defined by equality $\mathrm{V}_{m,n}(e)=\mathrm{U}_{m,n}\left(e;x^{m}y^{n}\right)$ and if $\phi(\theta)$ is quite small and tends towards zero quite quickly with $\theta$ we will still have $V_{m,n}(e)\neq 0$ .

It is therefore possible to define, under the conditions indicated, the difference divided by e by the ratio

[e;f]_{m,n}=\frac{\mathrm{U}_{m,n}(e;f)}{\mathrm{V}_{m,n}(e)}

Let's add to the set $\mathrm{E}_{1}^{*}$ all the limits of elements on which the divided difference can be defined in this way. We then see that if a sequence of elements of $\mathrm{E}_{1}^{*}$ tends towards such a limiting group $e$ , the corresponding divided differences have a limit $[e;f]_{m,n}$ .

Let us assume in particular that $f(x,y)$ either continuous and have a derivative $\frac{\partial^{m+n-2}f}{\partial x^{m-1}\partial y^{n-1}}$ continuous; therefore all derivatives $\frac{\partial^{r+s}f}{\partial x^{r}\partial y^{s}}r\leq m-1_{2}$ . $s\leq n-1$ exist and are continuous. We then know that the function has a normal divided difference of order ( $m,n$ ) which can be extended over any limit group containing at least four distinct points.

Let us further suppose that the derivative $\frac{\partial^{m+n-2}f}{\partial x^{m-1}\partial y^{n-1}}$ has a normal divided difference of order $(1,0)$ and a normal divided difference of order (0,1) bounded in E. By considerations analogous to those made above, it is shown that the function then has a normal divided difference of order ( $m,n$ ) which can be extended over any limit group e containing at least two distinct points. It should be noted that if ea contains only two distinct points, the value of $[e;f]_{m,n}$ may not be determined; we can only affirm that this quantity remains bounded.

Either $e^{(p)}p=\mathrm{J},2,\ldots$ a series of elements of $\mathrm{E}^{*}{}_{1}$ tending towards
( ⁵³ ) The rate at which the characteristic function must tend towards zero for $\theta\rightarrow 0$ generally depends on the function $f(x,y)$ This speed can be specified using the oscillation moduli of various orders of the function $f(x,y)$
boundary group $e$ of this nature. The following $\left[e^{(p)};f\right]_{m,n}p=1,2,\ldots$ does not necessarily tend towards a limit; we can only state that all limit values of this sequence are finite (the set of limit values being closed, it will necessarily be bounded).

We can now state the following property:
If the derivative $\frac{\partial^{m+n-2}f}{\partial x^{m-1}\partial y^{n-1}}$ of the function $f(x,y)$ exists and has a normal divided difference of order $(1,0)$ and a normal divided difference of order $(0,1)$ bounded in E. If, in addition, the function $f(x,y)$ has a normal divided difference of order ( $m,n$ bounded at every point ac E, it also has a normal divided difference of order ( $m,n$ ) boonée in E.

The function does indeed have a normal divided difference of order $(m,n)$ less extensive than the given one and which can be extended over any limit group having at most two distinct points. This normal divided difference is obviously bounded at every point of E. Suppose that it is not bounded in E and then consider a sequence of elements $e^{(p)}$ of $\mathrm{E}_{1}^{*}$ tending towards a borderline group $e$ such as $\left[e^{(p)};f\right]_{m,n}$ tends towards $+\infty$ We deduce that all points of $e$ must be identical. Therefore, there would exist a point where the normal divided difference under consideration is unbounded, which is impossible. The property is thus proven.

This property completes and generalizes some results of No. 40.
44. The extension on any limit group is possible, in the sense of the previous No., if the function is a polynomial and if the normal divided difference considered is regular.

Suppose that the function is a divided-intercept function bounded with respect to $x$ for any value of $y$ and to $n^{\text{ème }}$ difference divided bounded with respect to $y$ for any value of $x$ Consider a normal and regular divided difference of order ( $m,n$ ) of this function, having as its characteristic function $\phi(0)$ Suppose that this normal divided difference is not bounded at a point $M(x,y)$ of $E$ .

We will demonstrate, under these assumptions, that the normal divided difference of order ( $m+1,n$ ), of the same characteristic function $\phi(\theta)$ cannot be bovnée in E:

$1^{0}$ the points $\mathbb{M}_{i,j}^{(p)}\left(x_{i,j}^{(p)},y_{i,j}^{(p)}\right),1\leq i\leq m+1,1\leq j\leq n+1,x_{1,j}<x_{2,j}<\ldots\ldots<x_{m+1,1};y_{i,1}<y_{i,2}<\ldots<y_{i,n+1}$ of $e^{(p)}$ all tend towards M. $2^{0},\left[e^{(p)};f\right]_{m,n}$ tends towards $+\infty$ We are modifying $\mathrm{U}_{m,n}\left(e^{(p)};f\right)$ as in the previous No.; we therefore replace the line corresponding to the point $\mathrm{M}_{i,j}^{(p)}$ by (85) in which each term has been substituted for its divided difference of order ( $i-1,j-1$ ) taken on the points $M_{\alpha,\beta}^{(p)},1\leq\alpha\leq i$ , $1\leq\beta\leq\equiv j$ We perform the same operation on $\mathrm{V}_{m,n}\left(e^{(p)}\right)$ by replacing $f$ by $x^{m}y^{n}$ In this way the report $\left[e^{(p)};f\right]_{m,n}$ It doesn't change. Either now $e_{1}^{(p)}$ the group of $(m+2)(n+1)$ points formed by $e^{(\rho)}$ and by the points $\mathrm{P}_{1,j}^{(\rho)}\left(x_{0},y_{m+1,j}^{(\rho)}\right)1\leq j\leq n+1,x_{0}$ being fixed and $>x+\rho$ At the limit, the points $\mathrm{P}_{1,j}^{(p)}$ will merge with the point ( $x_{0},y$ ). We are also modifying the report $\left[e_{1}^{(p)};t\right]_{m+1,n}$ as above.

Let's do $p\rightarrow\infty$ , so either $\left|\left[e_{1}^{(p)};f\right]_{m+1,n}\right|$ has an infinite limit, or else it remains bounded. In the latter case, it is immediately apparent that at least one of the divided differences must

	$\displaystyle{\left[\mathrm{M}_{1,1}^{(p)},\mathrm{M}_{1,2}^{(p)},\ldots,\mathrm{M}_{i,j}^{(p)};t\right]_{i-1,j-1}}$		(86)
	$\displaystyle i=1,2,\ldots,m+1,\quad j=1,2,\ldots,n+1$

(the system $i=m+1,\quad j=n+1$ being excluded.)
or unbounded.
Suppose that (86) is unbounded for $i=r+1,j=s+1,r\leq m$ , $s\leq n,r+s<m+n$ and that the differences divided (86) for $i=1,2,\ldots,r+1,j=1,2,\ldots,s+1,i+j<r+s+2$ remain limited. One can always determine $r$ And $s$ in this way (if $r=s=0$ the function is unbounded at the point $M$ ).

Let us designate by $e_{2}^{(p)}$ the whole of $(m+2)(n+1)$ points

\left.\begin{array}[]{ll}\mathrm{M}_{i,j}^{(p)}&i=1,2,\ldots,r,j=1,2,\ldots,s\\ \left(x_{i,1}^{(p)},y_{j}\right)&i=1,2,\ldots,r+1,j=1,2,\ldots,n-s\\ \left(x_{i},y_{j}\right)&i=1,2,\ldots,m-r+1,j=1,2,\ldots,n-s\\ \left(x_{i},y_{r+1,j}^{(p)}\right)&i=1,2,\ldots,m-r+1,j=1,2,\ldots,s+1\end{array}\right\}\text{ (aucun si }n=s\text{ ) }

Or $x+p<x_{1}<x_{2}<\ldots<x_{n-r+1};y_{1}<y_{2}<\ldots<y_{n-s}<y-p$ are fixed x-coordinates and y-coordinates.

Doing $p\rightarrow\infty$ the difference divided $\left[e_{c}^{(p)};f\right]_{m+1,n}$ does not remain confined, which demonstrates ownership.

It is demonstrated in the same way that the normal divided difference of order ( $m,n+1$ ) and more generally than the normal divided difference of order $\left(m_{1},n_{1}\right)m_{1}\geq m,n_{1}\geq n,m_{1}+n_{1}>m+n$ , of the same characteristic function $\phi(\theta)$ , cannot be bounded in E.
45. Now suppose that the function $f(x,y)$ has a normal divided difference of order ( $m,n$ ) bounded in E. We will demonstrate that the partial divided difference of order ( $m,0$ ) and the partial divided difference of order ( $0,n$ ) are bounded in E. It suffices to show that these divided partial differences are bounded at every point of E. We obtain this property, for the order ( $m,0$ ) for example, by reasoning analogous to that used for the demonstration of the properties of No. 39 and by making use of the identity, easy to establish-

\left[\mathrm{M}_{1},\mathrm{M}_{2},\ldots,\mathrm{M}_{m+1};f\right]_{m,0}=

$=\sum_{i=1}^{n}(-1)^{i+1}\frac{\mathrm{\penalty 10000\ V}\left(y,y_{1}\ldots,y_{i-1},y_{i+1},\ldots,y_{n}\right)}{\mathrm{V}\left(y_{1},y_{2},\ldots,y_{n}\right)}\left[\mathrm{M}_{1,i},\mathrm{M}_{2,i},\ldots,\mathrm{M}_{m+1,i};f\right]_{m,0..}+\frac{(-1)^{n}\mathrm{\penalty 10000\ V}\left(y,y_{1}y_{2},\ldots,y_{n}\right)}{\mathrm{V}\left(y_{1},y_{2},\ldots,y_{n}\right)}\left[\mathrm{M}_{1},\mathrm{M}_{2},\ldots,\mathrm{M}_{m+1},\mathrm{M}_{1,1},\mathrm{M}_{1,2},\ldots,\mathrm{M}_{m+1,n};\right]_{m,n}$
Or, $\left(x^{\prime}{}_{i},y\right)$ are the coordinates of the point $\mathrm{M}_{i},i=1,2,\ldots,m+1$ And. ( $x_{i},y_{j}$ ) are the coordinates of the points $\mathrm{M}_{i,\mathrm{j}},i=1,2,\ldots,m+1,j=1,2,\ldots,n$ Here we assume that the points $\mathrm{M}_{i,\mathrm{j}}$ remain fixed and that $x_{i}^{\prime},y$ vary within the range permitted by the characteristic function of the difference: divided by the given.

The results of the previous issue show us that:
If a function $f(x,y)$ has a normal divided difference of order ( $m,n$ ) bounded in E, it also has a normal divided difference of order ( $m-1,n$ ) and a normal divided difference of order ( $m,n-1$ ) bounded in $\mathrm{E}_{\mathrm{o}\ldots}$

We deduce that the function also has a normal divided difference of order $(m,0)$ bounded in E. The derivative $\frac{\partial^{m-1}f}{\partial x^{m-1}}$ Therefore, it exists and is continuous. Now, consider the normal divided difference of order ( $m-1,n$ ) which is also bounded, and let us only consider the differences divided… taken from distributed points $m$ has $m$ on $n+1$ parallel to the axis $OX$ By taking a limit, we deduce that the derivative $f_{x^{m-1}}^{(m-1)}$ also has a normal divided difference of order ( $0,n$ ) bounded. It follows that:

If the function $f(x,y)$ has a normal divided difference of order ( $m,n$ ) bounded in E, it has a derivative $f_{x^{m-1}y^{n-1}}^{(m+n-2)}$ continues in all… point of E.

It can also be demonstrated more precisely that the derivative $f_{x^{\prime}y}^{(r+s)}r\leq m-1,s\leq n-1$ has a normal divided difference of order: ( $m-r,n-s$ ) bounded in E.

Suppose that a normal divided difference of order ( $m,n$ ), of the function $f(x,y)$ , such that the divided difference has a finite and well-defined limit when the points on which it is taken tend in any way towards a limit point. This limit is equal at every point to $\frac{1}{m!n!}f_{x^{m}y^{n}}^{(m+n)}$ The derivative $f_{x^{m}y^{n}}^{(m+n)}$ exists and is continuous in E. Of course, the limits in question only exist for a particular class of functions depending on the normal divided difference under consideration. This class includes polynomials if the normal divided difference is regular.

CHAPTER VI.

ON convex functions of two real variables.

S 1. - First extension of the notion of convexity.

46.

Partial divided differences allow an immediate generalization of the convexity of any order for functions of two independent variables.

The function $f(x,y)$ 's will be convex, non-concave, polynomial, non-convex or concave of order ( $m,n$ ) on the set E following that we have

\left[\begin{array}[]{l}x_{1},x_{2},\ldots,x_{m+2}\\ y_{1},y_{2},\ldots,y_{n+2}\end{array};f\right]>0,\geq 0,=0,\leq 0,<0,\text{ sur E. }

We can distinguish this kind of convexity by the designation of partial convexity, non-concavity, etc., but we suppress this distinction, it being understood that we only speak in § of this kind of convexity.

The set of functions defined above constitutes the class of (partial) order functions ( $m,n$ ).

The value $m=-1$ is not excluded. A function of order ( $-1,n$ ) is a function that enjoys the same convexity property of order $n$ relative to the variable $y$ , for any value da $x$ A function of order ( $m,-1$ Finally, the order functions $(-1,-1)$ are functions with an invariable sign.

A geometric definition, analogous to that given for functions of one variable, is obtained using pseudo-polynomials.

So indeed

\mathrm{P}\left(\begin{array}[]{l}x_{1},x_{2},\ldots,x_{m+1}\\ y_{1},y_{2},\ldots,y_{n+1}\end{array};f\binom{x}{y}\right.

pseudo-polynomial key of order ( $m,n$ ) taking the values of $f(x,y)$ on the network $x=x_{i},i=1,2,\ldots,m+1,y=y_{j},j=1,2,\ldots,n+1$ where we can assume that the sequences $x_{1},x_{2},\ldots,x_{m+1},y_{1},y_{2},\ldots,y_{n+1}$ be ordered. Formula (83) then shows us that non-concavity (convexity) is expressed by the fact that the function must be at every point of the rectangle

\begin{array}[]{ll}\left(x_{i},y_{\mathrm{j}}\right)&\left(x_{i},y_{\mathrm{j}+1}\right)\\ \left(x_{i+1},y_{\mathrm{j}}\right)&\left(x_{i+1},y_{\mathrm{j}+1}\right)\end{array}

inon below (above) or not above (below) the pseudopolynomial (88) depending on whether $n+m-i-j$ is even or odd. This property applies to the entire rectangle E, by agreeing to set in (89) $x_{0}=a,x_{m+2}=b,y_{o}=c,y_{n+2}=d$ .

Finally, let us say that we can also consider functions possessing several convexity properties and thus define various classes of functions, as in the case of a single variable.
47. The properties of convex functions of a single variable do not generalize entirely to this case. For example, a function of order ( $n,n$ ) in the closed rectangle (81), is not necessarily bounded. But:

If a function of order ( $m,n$ ) in rectangle E is bounded on a lattice of order $(m+2,n+2)$ It is bounded within the smallest rectangle containing the nodes of this network.

In particular, if the sides of E belong to the lattice, the function is bounded throughout the rectangle E.

We also see that:
If the function is of order ( $m,n$ ) and if we have

\left[\begin{array}[]{l}x_{1},x_{2},\ldots,x_{m+2}\\ y_{1},y_{2},\ldots,y_{n+2}\end{array};\mathrm{f}\right]=0

it is polynomial, therefore reduces to a pseudo-polynomial of order ( $m,n$ ) in the smallest rectangle containing the points ( $x_{i},y_{j}$ ).

The demonstration is immediate.
48. Consider $pq(p\geq m+2,q\geq n+2)$ points $\mathrm{M}\left(x_{i},y_{j}\right)i=1,2,\ldots,p$ , $j=1,2,\ldots,q$ , in rectangle E and suppose that the sequences $x_{1},x_{2},\ldots,x_{p}$ ; $y_{1},y_{2},\ldots,y_{q}$ are ordered. We then see, as in the case of a single variable, that the necessary and sufficient conditions for non-concavity (convexity) of order ( $m,n$ ) on the points are

\begin{gathered}{\left[\begin{array}[]{l}x_{l},x_{i+1},\ldots,x_{i+m+1}\\ y_{j},y_{j+1}y\ldots,y_{j+n+1};f\end{array}\right]\geq 0,(>0)}\\ i=1,2,\ldots,p-m-1,j=1,2,\ldots,q-n-1.\end{gathered}

Formula (1) shows us that the sequences

		$\displaystyle{\left[\begin{array}[]{l}x_{1},x_{2},\ldots,x_{m+1}\\ y_{j},y_{j+1},\ldots,y_{i+n+1}\end{array};f\right],\left[\begin{array}[]{l}x_{2},x_{3},\ldots,x_{m+2}\\ y_{j},y_{j+1},\ldots,y_{j+n+1}\end{array};f\right],\ldots}$
		$\displaystyle{\left[\begin{array}[]{l}x_{i},x_{i+1},\ldots,x_{i+m+1}\\ y_{1},y_{2},\ldots,y_{n+1}\end{array};f\right],\left[\begin{array}[]{l}x_{i},x_{i+1},\ldots,x_{i+m+1}\\ y_{2},y_{3},\ldots,y_{n+2}\end{array};f\right],\ldots}$

are ordered. It is easy to deduce that:
If the function is of order ( $m,n$ ) in rectangle E and if the partial divided differences of order ( $m+1,n$ ), ( $m,n+1$ ) are bounded: on a network of order $(2m+2,2n+2)x=x_{i},i=1,2,\ldots,2m+2$ , $y=y_{j},j=1,2,\ldots,2n+2$ , these divided differences will be bounded within: the entire rectangle

\begin{array}[]{ll}\left(x_{m+1},y_{n+1}\right)&\left(x_{m+1},y_{n+2}\right)\\ \left(x_{m+2},y_{n+1}\right)&\left(x_{m+2},y_{n+2}\right)\end{array}

If, in addition, the partial divided differences of order ( $m+1,0$ ). And ( $0,n+1$ ) are bounded on a network of order ( $m+1,n+1$ ), included in rectangle (90), all the partial divided differences of order $<(m+1,n+1)$ are bounded within this rectangle. We demonstrate: again, exactly as in No. 13 for functions of one variable, that:

There are functions of a given class that exist in advance of the $pq$ points considered provided that if ( $m,n$ ) is the highest-order polynomiality; all higher-order properties are polynomiality.

It should be noted that here a polynomial function of order $(m,n)$ is not necessarily of order ( $m-1,n$ ) Or ( $m,n-1$
49. We have the following property :

If a function is of order $(1,-1)$ and order $(-1,1)$ It is continuous with respect to the set of variables at every internal point…

This property was demonstrated by MP Montel by assuming that the function is non-concave of order $(1,-1)$ and non-concave: of order $(-1,1)\left({}^{54}\right)$ .
M. N. Kritikos a généralisé la propriété précédente de la manière suivante : ( ⁵⁵ )
(54) P. Montel „Sur les fonctions doublement convexes et les fonctions doublement sous-harmoniques" Praktika de l’Acad. d’Athènes 6, (1931), p. 374. Une telle fonction est dite doublement convexe.
(55) N. Kritikos „Sur les fonctions multiplement convexes ou concaves ^∗∗. Praktiza de l’Acad. d’Athènes, 7 1932, p. 44. Voir aussi un mémoire demême auteur paru dans le Bulletin de la Soc. Math. de Grèce, t. XI (1930) pp. 21-28.

Si $f(x,y)$ est d’ordre 1 par rapport à l’une des variables et continue par rapport à l’autre, elle est continue par rapport à l’ensemble des variables en tout point interieur.

Plus généralement, si la fonction est d’ordre $n$ par rapport à l’une des variables et continue par rapport à l’autre elle est continue par rapport à l’ensemble des variables. Cette propriété peut d’ailleurs se déduire du théorème de M. Kritikos, compte tenant des résultats du No. 14 de la première partie.

Une fonction d’ordre $(m,-1)$ continue par rapport à $y$ dans le rectangle E a une différence divisée partielle d’ordre ( $m,0$ ) bornée dans tout rectangle complètement intérieur.

Soint en effet $\mathrm{E}\left(a^{\prime}\leq x\leq b^{\prime},c^{\prime}\leq y\leq d^{\prime}\right),a<a^{\prime}<b^{\prime}<b$ , $c<c^{\prime}<d^{\prime}<d$ un rectangle complètement intérieur. Prenons les abscisses $x_{1}^{\prime},x_{2}^{\prime},\ldots,x_{m+1}^{\prime}$ à l’intérieur de ( $a,a$ ) et les abscisses $x^{\prime\prime}{}_{1},x^{\prime\prime}{}_{2},\ldots,x^{\prime\prime}{}_{m+1}$ à l’intérieur de ( $b^{\prime},b$ ). Nous savons (No. 17 de la première partie) que si $x_{1},x_{2},\ldots,x_{m+1}$ sont dans l’intervalle fermé ( $a^{\prime},b^{\prime}$ ), la différence divisée

\left[\begin{array}[]{l}x_{1},x_{2},\ldots,x_{m+1};f\\ y;\end{array}\right]

est comprise entre les difiérences divisées

\left[\begin{array}[]{l}x_{1}^{\prime},x_{2}^{\prime},\ldots,x_{m+1}^{\prime}\\ y\end{array};f\right],\left[\begin{array}[]{l}x_{1}^{\prime\prime},x_{2}^{\prime\prime},\ldots,x_{m+1}^{\prime\prime}\\ y\end{array};f\right]

Or, la fonction étant continue par rapport a $y$ elle est bornée sur l’ensemble des parallèles $x=x_{i}^{\prime},x=x_{i}^{\prime\prime}i=1,2,\ldots,m+1$ . ll en résulte que les différences divisées (91) restent bornées dans leur ensemble lorsque $y$ varie, se qui démontre la propriété.

En particulier une fonction qui est d’ordre ( $m,-1$ ) et d’ordre $(-1,n)$ a une différence divisée partielle d’ordre ( $m,0$ ) et une différence divisée partielle d’ordre ( $0,n$ ) bornées dans tout rectangle complètement intérieur.

Tenant compte d’un théorème de M. P. Montel (56) on en déduit la propriété :

Une fonction qui est d’ordre ( $m,-1$ ) et d’ordre ( $-1,n$ ) a en tout point intérieur une dérivée $\frac{\partial^{r+s}f}{\partial x^{r}\partial y^{s}}$ continue par rapport à l’ensemble des variables, pourvu que

\frac{r}{m}+\frac{s}{n}<1

(56) P. Montel loc, cit. (35).

Mathemalica VIH.

Si la fonction est d’ordre ( $m,n$ ) at si la dérivée partielle $f^{\prime}x$ existe c’est une fonction d’ordre ( $m-1,n$ ) présentant le même caractère de convexité et réciproquement. Plus généralement si $t$ est d’ordre ( $m,n$ ) et si $f_{xy}^{(r+s)}{}^{s}$ existe c’est une fonction d’ordre ( $m-r,n-s$ ). Si $f_{x}^{(m+1,n+2)}y^{n+1}$ existe, elle est non négative si la fonction est non-concave d’ordre ( $m,n$ ) et réciproquement. On suppose ici encore que E soit un rectangle.
50. Avant de tinir ce § disons qu’on peut définir la convexité avec d’autres différences divisées que les différences divisées parielles.

On peut par exemple donner des définitions à l’aide des différences divisées normales. Il est inutile de répéter comment on écrit ces conditions. Ces fonctions jouissent des propriétés plus restrictives que celles précédemment définies. Considérons par exemple une fonction qui est d’ordre ( $m,-1$ ) par rapport à une différence divisée normale d’ordre ( $m+1,0$ ). Une telle fonction a une différence divisée normale d’ordre ( $m,0$ ) bornée dans tout rectangle complètement intérieur. Elle a donc des dérivées partielles $\frac{\partial^{i}t}{\partial x^{i}}i=1,2,\ldots,m-1$ continues en tout point intérieur. La dérivée $f^{\prime}x$ est d’ailleurs a son tour d’ordre ( $m-1,-1$ ) par rapport à une certaine différence divisée normale d’ordre ( $m,0$ ). Il est à remarquer que si la dérivée ${\underset{x}{f}}_{m+1}^{(m+1)}$ existe elle est d’un signe invariable, mais la récjproque n’est pas vraie. Il faut des conditions supplémentaires de continuité pour pouvoir affirmer que de l’inégalité $f_{\begin{subarray}{c}m+1\\ m+1\end{subarray}}^{(m+1)}\geq 0$ résulte la non-concavité d’ordre $(m,-1)$ de la fonction par rapport à une différence divisée normale d’ordre ( $m+1,0$ ).

§ 2. - Seconde extension de la notion de convexité.

51.

Considérons une fonction $f(x,y)$ définie, pour ne pas compliquer, sur un domaine fermé convexe et borné E. L’allure de la fonction sur une droite s’obtient en prenant le plan perpendiculaire sur XOY qui se projète sujvant cette droite et en considérant la fonction dans ce plan. L’axe OY dans ce plan est orientée vers le OZ positif.

Nous nous proposons d’étudier les fonctions qui sont d’ordre $n$ sur toute droite contenant des points de E. Nous dirons d’une telle fonction qu’elle est d’irdre $n$ sur l’ensemble E.

Supposons que $n$ soit pair. Nous avons vu que le caractère de convexité d’une fonction d’une variable dépend de l’orientation de l’axe OX. Pour cette raison nous ne ferons pas de distinction entre lacon-
wexité et la concavité, resp. entre la non-concavité et la non-convexité sur une droite. Une fonction peut être d’ordre $n$ au sens strict ou au sens large suivant qu’elle est convexe (ou concave) resp. non-concave (ou non-convexe). Elle peut enfin être polynomiale sur une droite.

Supposons maintenant que $n$ soit impair. La nature de convexité d’une fonction d’une variable et d’ordre impair ze dépend pas de l’orientation de l’axe OX. On peut donc ici faire la distinction entre la convexité, non-concavité, polynomialité, non-convexité et la concavité sur une droite. Une fonction d’ordre impair $n$ sera dite convexe, nonconcave, …. etc. d’orde $n$ sur E si elle est convexe, non-concave, … etc. d’ordre $n$ sur toute droite de E.

On peut distinguer la sorte de convexité ainsi introduite en disant qu’il s’agit d’une convexité, non-concavité …. etc. totale d’ordre $n$ . Nous supprimons dans la suite cette dénomination étant sous endendu qu’il no s’agira que de cette sorte de convexité.

Considérons par exemple un polynome de degré $n+1$

a_{0}x^{n+1}+a_{1}x^{n}y+\ldots.+a_{n+1}y^{n+1}+\ldots

C’est toujours une fonction d’ordre $n$ . Si $n$ est pair il y a toujours des droites sur lesquelles la fonction est polynomiale. Si $n$ est impair et si le polynome $a_{0}t^{n+1}+a_{1}t^{n}+\cdots+a_{n+1}$ est positif (non négatif) la fonction est convexe (non-concave) d’ordre $n$ . Sur cet exemple on voit bien qu’une fonction peut être d’ordre $n$ sans présenter un caractère de convexité déterminé.

On pourait également considérer des classes de fonctions présentant plusieurs propriétés de convexité déterminées.
52. Je dis que si la fonction est d’ordre $n(n>0)$ elle présente le même caractère de convexité sur des droites parallèles.

On peut supposer que les droites soient parallèles à l’axe OX. La démonstration se fait alors très facilement en tenant compte du sfait que si la suite d’ordonnées $y_{1},y_{2},\ldots,y_{p},\ldots$ tend vers l’ordonnée $y_{0}$ , la suite de fonctions de $x,f\left(x,y_{1}\right),f\left(x,y_{2}\right),\ldots,f\left(x,y_{p}\right),\ldots$ converge vers $f\left(x,y_{0}\right)$ .

Une fonction d’ordre $n(n>0)$ est en particulier d’ordre $n$ par rapport à chacune des variables ; elle est donc d’ordre ( $n,-1$ ) et d’ordre $(-1,n)$ . Nous en déduisons que :

Toute fonction d’ordre $n$ sur E est continue en tout point intérieur $-sin>0$ .

En particulier :
Toute fonction d’ordre $n$ sur $\mathbb{E}$ est bornée dans tout domaine comvolètement intérieur à E .

Je dis que cette propriété est vraie même pour $n=0$ . Soit $\left\{\mathrm{M}_{\rho}\right\}$ une suite de points de E tendant vers un point limite intérieur M. Prenons un point $M^{\prime}$ et une droite $\Delta$ dans $E$ de manière que pour $p>\mathrm{N}$ l’intersection des droites $\mathrm{M}^{\prime}\mathrm{M}_{p}$ et $\Delta$ tombe à l’intérieur de E . Designons par $\mathrm{M}^{\prime}{}_{p}$ l’intersection des droites $\mathrm{M}^{\prime}\mathrm{M}_{p}$ et $\Delta$ . La valeur de la fonction en $\mathrm{M}_{p}$ est toujours comprise entre ses valeurs en $\mathrm{M}^{\prime}$ et $\mathrm{M}_{p}^{\prime}$ . On voit maintenant que si nous supposons que les valeurs de la fonction aux points $M_{p}$ aient une limite infinie il arrive ou bien qu’en M’ la fonction ne soit pas bornée ou bien qu’elle ne soit pas bornéesur $\Delta$ au voisinage de l’intersection de cette droite avec MM’, ce qui est impossible. La propriété est donc démontrée.
53. Nous avons encore la propriété suivante :

Une fonction d’ordre $n$ dans le domaine $\mathbf{E}$ a des dérivécs partielles d’ordre $<n$ continues en tout point intérieur.

En ce qui concerne les dérivées d’ordre $n$ , nous savons qu’en toutr point interieur elles existent suivant toute demi-droite issue de ce point.

Soient $M_{1}\left(x_{1},y_{1}\right),M_{2}\left(x_{2},y_{2}\right)$ deux points et prenons la différencedivisée première sur la droite joignant ces deux points. Nous avons-

\frac{f\left(x_{1},y_{1}\right)-f\left(x_{2},y_{2}\right)}{M_{1}-M_{2}}=\cos\alpha\cdot\left[x_{1},x_{2}f\left(x,y_{1}\right)\right]+\sin\alpha\cdot\left[y_{1},y_{2}f\left(x_{2},y\right)\right]

où $\alpha$ est l’angle de la droite $\mathrm{M}_{1}\mathrm{M}_{2}$ avec l’axe OX .
Nous en déduisons facilement que la différence divisée d’ordro $n+1$ sur $n+2$ points $\mathrm{M}_{i}\left(x_{i},y_{i}\right),i=1,2,\ldots,n+2$ en ligne droite s’écrit

\sum_{i=0}^{n+1}\cos^{n+1-i}\alpha\sin^{i}\alpha\cdot\left[\begin{array}[]{l}x_{i+1},x_{i+2},\ldots,x_{n+2}\\ y_{1},y_{2},\ldots,y_{i+1}\end{array};f\right]

Si la fonction est d’ordre $n$ cette expression est de signe invariable sur toute droite.

Supposons en particulier que les dérivées d’ordre $n+1$ existent. Il faut alors et il suffit que la fonction

\sum_{i=0}^{n+1}\binom{n+1}{i}\cos^{n-i+1}\alpha\sin^{i}\alpha\cdot\frac{\partial^{n+1}T}{\partial x^{n-i+1}\partial y^{i}}

(92)

soit de signe invariable sur toute droite faisant l’angle $\alpha$ avec l’axe $OX$ .
Si la fonction est d’ordre impair et convexe (non-concave) lepolynome est non négatif en tout point où les dérivées existent. Reciproquement, si les dérivées existent et si en tout point intérieur le polynome (92) est non négatif resp. positif on peut affirmer quela fonction est non-concave resp. convexe d’ordre impair $n$ . Si $n=1$ , pour que la fonction soit non-concave d’ordre 1 il faut et il
suffit que $f^{\prime\prime\prime}{}_{x^{2}}\geq 0,f^{\prime\prime}{}_{x^{2}}f^{\prime\prime}{}_{y^{2}}-\left(f^{\prime\prime}{}_{xy}\right)^{2}\geq 0$ et pour qu’elle soit convexe il suffit que $f^{\prime\prime}{}_{x^{2}}>0,f^{\prime\prime}{}_{x^{2}}f^{\prime\prime}{}_{y^{2}}-\left(f^{\prime\prime}{}_{xy}\right)^{2}>0$ assuming, of course, that second derivatives exist.

First-order functions are very close to the convex functions of Mr. Jensen ( ⁵⁶ ). Mr. Jensen defines a convex function by the inequality

f\left(\frac{x+x^{\prime}}{2},\frac{y+y^{\prime}}{2}\right)\leq\frac{1}{2}\left(f(x,y)+f\left(x^{\prime},y^{\prime}\right)\right)

His points $(x,y),\left(x^{\prime},y^{\prime}\right),\left(\frac{x+x^{\prime}}{2},\frac{y+y^{\prime}}{2}\right)$ belonging to the domain of the function. If such a function is bounded, it is non-concave of order 1, according to our definition.
54. - A polynomial function reduces to a polynomial on any line. We will demonstrate that:

A polynomial function of order $n$ reduces on E to the variables of a polynomial of degree n.

Let's take in E the $\frac{1}{2}(n+1)(n+2)$ points $\mathrm{M}_{l,\mathrm{j}}\left(x_{i},y_{j}\right)i=1,2,\ldots\ldots,j,j=1,2,\ldots,n+1$ and consider the polynomial of degree $n$ taking the values $f\left(x_{i},y_{i}\right)$ to the points $\mathrm{M}_{i,\mathrm{j}}$ This polynomial is well-defined. It suffices to pass through a point $M$ of $E$ a suitable line and apply the polynomial property on this line to see that the function takes the same value at M as this polynomial.

Several other observations can be made about polynomial functions. For example, if the function is of order $n$ and coincides with a polynomial of degree $n$ on a certain number of line segments, it is also polynomial on any segment that has its endpoints on the segments considered and furthermore contains at least $n$ other points belonging to the given segments.

If the function is of order $n$ and if it reduces to the same polynomial of degree $n$ on the segments $\mathrm{AB},\mathrm{AA}_{i},\mathrm{BB}_{i},i=1,2,\ldots,n-1\mathrm{AC},\mathrm{BC}$ , the points $\mathrm{A}_{1},\mathrm{\penalty 10000\ A}_{2},\ldots,\mathrm{\penalty 10000\ A}_{n-1}$ being on the BC segment and $\mathrm{B}_{1},\mathrm{\penalty 10000\ B}_{2},\ldots,\mathrm{\penalty 10000\ B}_{n-1}$ on segment AC, it is polynomial of order $n$ in triangle ABC. To see that any segment, having its endpoints on the sides of the triangle, still contains at least $n$ For these points, it suffices to complete the given segments with those parallel to sides AC and BG. These parallels obviously contain $n$ points belonging to the given segments.
(56) See JLW Jensen loc. cit. (42).

TABLE OF CONTENTS.

M. Biernacki. On the cubic equation ….. 196
J. Capoulade. On certain second-order and elliptic type partial differential equations with singular coefficients ….. 139
J. Chazy. The mathematical work of Painlevé ….. 201
J. Devisme. On the partial differential equations of Messrs. P. Humbert and M. Ghermanesco ….. 147
RH Germay. Essay on the principle of virtual work ….. 126
M. Ghermanesco. On the equation $\Delta^{n}u=0$ ….. 134
C. Jacob. On a problem concerning gas jets ….. 205
Tib. Popoviciu. On some properties of functions of one or two real variables ….. 1
J. Rudnicki. Remark on a theorem of M. Walsh ….. 136
W. Sierpinski. On sets always of the first category ….. 191
W. Slebodzinski. On tensor differential forms and Poincaré's theorem ….. 86
JL Walsh. Note on the location of the roots of the derivative of a polynomial ….. 185
T. Wazewski. On an integral problem relating to the equation $\frac{\partial z}{\partial x}+Q(x,y)\frac{\partial z}{\partial y}=0$ ….. 103
EA Weiss. Zykliden als Bilder von Flächen 2. Ordnung in der Geraden-Kugeltransformation ….. 98
Editorial notes. ….. 212
Errata. ….. 212
Table of contents. ….. 213

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ON SOME PROPERTIES OF THE FUNCTIONS OF ONE OR TWO REAL VARIABLES.

Introduction.

Part one.

CHAPTER 1.

§ 1. - Bounded divided difference functions and functions

S. 2. - Properties of functions whose nth difference divided

§ 3. Propriétés des fonctions à nème variation bornée.

CHAPTER II.

§ 2. - Relations entre les fonctions d’ordre et de classe donnés et les fonctions étudiées au Chap. I.

CHAPTER III.

§ 1. - Some remarks on the definition of the derivative of ordernn.

§. 2. - Remarques sur les dérivées des fonctions étudiées aux Chap. I. et II.

Posons

§ 3. - On the limitation of the derivative of a function of ordernn.

Chapter IV.

SECONDE PARTIE.

SUR LES FONCTIONS CONVEXES D’ORDRE SUPÉRIEUR DE DEUX VARIABLES RÉELLES.

CHAPITRE V.

§ 1. - Théorie générale des différences divisées.

§. 2. - On a particular divided difference.

S. 3. - Study of another particular divided difference.

CHAPTER VI.

S 1. - First extension of the notion of convexity.

§ 2. - Seconde extension de la notion de convexité.

TABLE OF CONTENTS.

Related Posts

§ 1. - Some remarks on the definition
of the derivative of order $n$ .

§ 3. - On the limitation of the derivative of a function of order $n$ .