ON BOUNDED SOLUTIONS AND MEASURABLE SOLUTIONS OF CERTAIN FUNCTIONAL EQUATIONS

Tiberiu Popoviciu
in Cernăuţi

Received November 5, 1937.

SUMMARY

INTRODUCTION.
CHAPTER I. - Notations and some preliminary properties.
CHAPTER II. - On a class of functional equations in one variable.
CHAPTER III. - On pseudo-polynomials of two or more variables.
CHAPTER IV. - On some functional equations in several independent variables.
CHAPTER V. - On some functional properties characterizing polynomials of two variables.
BIBLIOGRAPHY.

Introduction

The aim of this work is to solve the functional equation
(l) $\Sigma a_{i_{1}i_{2}\ldots i_{m}}f\left(x_{1}+\alpha_{1i_{1}}h_{1},x_{2}+\alpha_{2i_{2}}h_{2},\ldots,x_{m}+\alpha_{mi_{m}}h_{m}\right)=0$ ,
Or $f\left(x_{1},x_{2},\ldots,x_{m}\right)$ is the unknown function at $m$ variables. We assume that equation (I) is verified, in a certain domain D, whatever $x_{1},x_{2},\ldots,x_{m},h_{1},h_{2},\ldots,h_{m}$ . THE $a_{i_{1}i_{2}\ldots i_{m}}$ and the $\alpha_{ii_{j}}$ are given constants and the summation is extended to the values $i_{j}=0,1,\ldots,n_{j},j=1,2,\ldots,m$ .

We have already studied a particular case of equation (I), when there are two independent variables [11] (*).

⁰⁰footnotetext:(*) Bold numbers in brackets refer to the bibliography placed at the end of the work.

In the case of a variable, the equation can be written

\sum_{i=0}^{n}a_{i}f\left(x+\alpha_{i}h\right)=0

(II)

and is a generalization of the well-known difference equation,

\Delta_{h}^{n}f(x)=\sum_{i=0}^{n}(-1)^{ni}\binom{n}{i}f(x+ih)=0

(III)

In the case of $m$ variables, the difference equation can be written

\Delta_{h_{1}}^{n_{1}}\Delta_{h_{2}}^{n_{2}}\ldots\Delta_{h_{m}}^{n_{m}}f\left(x_{1},x_{2},\ldots,x_{m}\right),

(IV)

Or $\Delta_{h_{j}}^{n_{j}}$ operates on the variable $x_{j}$ .
Note that if the solution $f\left(x_{1},x_{2},\ldots,x_{m}\right)$ , of equation (I), admits a sufficient number of partial derivatives, it must verify a certain system of partial differential equations, homogeneous and with constant coefficients. These equations are easily obtained by making the $h$ in the first member of the equation. (I).

The general solution to this system of partial differential equations is a sum of functions of the form

x_{j_{1}}^{l_{1}}x_{j_{2}}^{l_{2}}\ldots{}_{j_{r}}^{l_{r}}\mathbf{A}\left(x_{j_{r+1}},x_{j_{r+2}},\ldots,x_{j_{m}}\right)

(V)

where the $L$ are non-negative integers and A is an arbitrary function of $mr$ variables (we have $r\geq 1$ ).

We demonstrate that the general solution to equation (I) is of the same form, under much more general assumptions. This is so if we assume the function to be measurable with respect to each of the variables. In particular, this is so if we assume the function B to be measurable.

We also report equations for which we have the same result under the sole assumption that the function is bounded. In particular, equation (IV) enjoys this property, as demonstrated (for $m=1.2$ ) MA Marchaud [8].

We have divided this work into five chapters.
In chap. I we study, in detail, the properties of the operation expressed by the first member of equation (I). These are algebraic properties which allow us to reduce the problem to the resolution of an equation of the same form but simpler
[equation (52) of chap. IV]. We can also say that we establish algebraic properties which allow us to always return to equations in which the coefficients $a_{i_{1}i_{2}}\ldots i_{m}$ have simpler values (are also equal to $\pm 1$ ).

In chap. II we make a complete study of equation (II), completing and generalizing our previous results [11].

Chapter III is devoted to the study of pseudo-polynomials of two or more independent variables. This preliminary study is necessary to be able to specify the form of the general solution of equation (I).

The problem of solving equation (I) is treated in chap. IV. We completely determine all solutions of the form (V). In particular, we establish necessary and sufficient conditions so that, under the indicated hypotheses, the general solution of the equation is a polynomial.

In Chapter V we make an application of the previous results. We demonstrate that, under very general hypotheses, any function of two variables, which is a pseudo-polynomial with respect to two completely distinct systems of axes, necessarily reduces to a polynomial. We also possess the generalization of this property for the case of $m$ variables, but our demonstration is based on the theory of equations that we call of the first kind. We have only pointed out these equations, their study will be the subject of another work and we will then also cover the generalization of the results of chap. V.

CHAPTER I

Notations and some preliminary properties

1.

Given a function $f(x)$ , of a variable $x$ , we define the operation $\Delta_{h}^{\left(\alpha_{i}\right)}$ by the following formula:

\Delta_{h}^{\left(\alpha_{i}\right)}f(x)=\sum_{i=0}^{n}a_{i}f\left(x+\alpha_{i}\dot{h}^{\prime}\right)

Such an operation is therefore characterized by two series of constants: the (real) coefficients $a_{0},a_{1},\ldots,a_{n}$ and the (real) pseudoperiods $\alpha_{0},\alpha_{1},\ldots,\alpha_{n}$ . In the problems that we will examine
in this work, each of these sequences has a homogeneous character; only the mutual relations of their terms intervene in an essential way. Of course, the operation $\Delta_{h}^{\left(a_{i}\right)}$ only makes sense if the coefficients $a_{i}$ are not all zero. Moreover, we will assume, in general, that all $a_{i}$ are different from zero and that the $\alpha_{i}$ are distinct. This being the case, the product of two operations $\Delta_{h}^{\left(\alpha_{i}\right)},\Delta_{h}^{\left(\beta_{i}\right)}$ is still an operation of the same nature. If the expressions $\Delta_{h}^{\left(a_{i}\right)}f(x)$ And $\Delta_{h}^{\left(\beta_{i}\right)}f(x)$ contain $n+1$ And $m+1$ terms respectively, the product expression $\Delta_{h}^{\left(\beta_{i}\right)}\Delta_{h}^{\left(a_{i}\right)}f(x)$ contains $(n+1)(m+1)$ terms in general, but this number can also be smaller. What is essential is that:

The product of two operations always has a meaning and this multiplication is commutative.

Let us examine some special cases of operation (1). The expression

\Delta_{h}^{n}f(x)=\sum_{i=0}^{n}(-1)^{ni}\binom{n}{i}f(x+ih)

(2)

is a difference of order $n$ of the function $f(x)$ . Let us pose $\phi(x)==\left(x-x_{0}\right)\left(x-x_{1}\right)\ldots\left(x-x_{n}\right)$ and suppose that $a_{i}=\frac{1}{\phi^{\prime}\left(x_{i}\right)}$ ; the corresponding operation then gives us the expression

\sum_{i=0}^{n}\frac{f\left(x+\alpha_{i}h\right)}{\phi^{\prime}\left(\alpha_{i}\right)}

(3)

This is a generalization of the difference (2). This expression has been studied, in particular, by MA Denjoy [5]. Another important special case is the case where $\alpha_{i}=i,i=0,1,\ldots,n$ . We obtain the expression

\Delta_{h}^{(n)}f(x)=\sum_{i=0}^{n}a_{i}f^{\prime}(x+ih)

(4)

where it is unnecessary to make the restriction $a_{i}\neq 0$ for all $i$ .
This is the case where the mutual ratios of the pseudo-periods $\alpha_{i}$ are rational. Finally, we also consider expressions of the form

\delta_{h}^{\left(\alpha_{i}\right)}f(x)=\Sigma(-1)^{i_{1}+i_{2}+\cdots+i_{n}}f\left[x+\left(i_{1}\alpha_{1}+i_{2}\alpha_{2}+\cdots+i_{n}\alpha_{n}\right)h\right],

(5)

where the summation is extended to all values $i_{j}=0,1$ , $j=1,2,\ldots,n$ .

Before going any further, let us say, once and for all, that we will speak of order, characteristic polynomial, reducibility, etc., regardless of the operation $\Delta_{h}^{\left(\alpha_{i}\right)}$ , of the expression $\Delta_{h}^{\left(\alpha_{i}\right)}f(x)$ , from the equation $\Delta_{h}^{\left(a_{i}\right)}f(x)=0$ ,… etc. The same common language will be used in the case of several variables.
2. Let us now define the order of expression (1). This order is equal to the number $k$ for which

\sum_{i=0}^{n}a_{i}=\sum_{i=0}^{n}a_{i}\alpha_{i}=\ldots=\sum_{i=0}^{n}a_{i}\alpha_{i}^{k-1}=0,\sum_{i=0}^{n}a_{i}\alpha_{i}^{k}\neq 0.

If $\sum_{i=0}^{n}a_{i}\neq 0$ the expression is of order 0. If $k=n$ the expression is of order $n$ and is necessarily of the form (3).

Let us attach to expression (1) the characteristic polynomial of the first type $\mathrm{F}^{*}(x)=\sum_{i=0}^{n}a_{i}x^{\alpha_{i}}$ . In general it is not a polynomial properly speaking, but a polynomial in $x^{\alpha_{i}}$ . The order $k$ is then characterized by the relationships

F^{*}(1)=F^{*\prime}(1)=\cdots=F^{*}(k-1)(1)=0,F^{*}(k)(1)\neq 0.

For example, expression (5) is of order $n$ .
Two expressions having characteristic polynomials $\mathrm{F}^{*}(x)$ And $F^{*}\left(x^{p}\right)$ , Or $p$ is a positive integer, are equivalent. In the case of expression (4) $\mathrm{F}^{*}(x)$ is indeed a polynomial. In particular, for (2) we have $\mathrm{F}^{*}(x)=(x-1)^{n}$ .

We will say that an expression of the form (1) is a consequent of (1) if its characteristic polynomial of the first type is of the form $\phi_{1}(x)F^{*}\left(x^{p_{1}}\right)+\phi_{2}(x)F^{*}\left(x^{p_{2}}\right)+\ldots$ Or $p_{1},p_{2},\ldots$ are positive integers $\phi_{1}(x),\phi_{2}(x),\ldots$ polynomials in $x$ or, more generally, linear combinations of certain powers of $x$ .

We also introduce a characteristic polynomial of the second type. We can always assume, without restricting the generality
, that in (1) we have $z_{0}=0$ . Let us then call expressión assbecieè i (1) any expression of the form

\Sigma bf\left(x+\beta_{i}h\right)

(6)

Or $b$ are the coefficients and the $\beta_{i}$ are of the form $r_{1}\alpha_{1}+r_{2}\alpha_{2}+\ldots\ldots+r_{n}\alpha_{n}$ , THE $r_{i}$ being positive or zero integers. We will then say that the polynomial at $n$ variables $\Sigma bx_{1}^{r_{1}}x_{2}^{r_{2}}\ldots x_{n}^{r_{n}}$ is the characteristic polynomial, of the second type, of expression (6). In this way, the characteristic polynomial of the second type is defined for all expressions associated with (1). In particular, expression (1) itself has the characteristic polynomial $\mathrm{F}\left(x_{1},x_{2},\ldots,x_{n}\right)=a_{0}+a_{1}x_{1}+\cdots+a_{n}x_{n}$ . Two expressions having characteristic polynomials $\mathrm{F}\left(x_{1},x_{2},\ldots,x_{n}\right),\mathrm{F}\left(x_{1}^{p},x_{2}^{p},\ldots,x_{n}^{p}\right)$ Or $p$ is a positive integer, are equivalent. Any expression (6) whose characteristic polynomial is of the form

\phi_{1}\mathrm{\penalty 10000\ F}\left(\lambda_{1}^{p_{1}},x_{2}^{p_{1}},\ldots,x_{n}^{p_{1}}\right)+\phi_{2}\mathrm{\penalty 10000\ F}\left(\lambda_{1}^{p_{2}},\lambda_{2}^{p_{2}},\ldots,x_{n}^{p_{3}}\right)+\ldots,

Or $p_{1},p_{2},\ldots$ are positive integers and $\phi_{1},\phi_{2},\ldots$ polynomials in $x_{1},x_{2},\ldots,x_{n}$ , is a consequent of (1).

In expression (1) $x$ And $h$ play the role of variables. It follows that expressions whose characteristic polynomial of the first type is of the form $x^{\beta}F^{*}(x)$ or those whose characteristic polynomial of the second type is of the form $x_{1}^{h}x_{2}^{l}\ldots x_{n}^{l}\mathrm{\penalty 10000\ F}\left(x_{1},x_{2},\ldots,x_{n}\right)$ are equivalent to (1).

This notion of equivalent expression is quite clear. Two expressions equivalent to a third are equivalent to each other. Any expression equivalent to a consequent of (1) is still a consequent of (1).

Consideration of characteristic polynomials considerably facilitates our study.

For example, expression (5), associated with (1), has the characteristic polynomial $\left(1-x_{1}\right)\left(1-x_{2}\right)\ldots\left(1-x_{n}\right)$ .
3. - We will now demonstrate that:

Every expression (1) has a consequent of the form (5).
Let us set $p=\left[\frac{n+1}{2}\right]$ , designating, as usual, by
[ $\alpha$ ] the largest integer included in $\alpha$ . Either
$\mathrm{F}_{i}=\mathrm{F}\left(x_{1}^{i},x_{2}^{i},\ldots,x_{n}^{i}\right)-\left(a_{p+1}x_{p+1}+a_{p+2}x_{p+2}+\ldots+a_{n}x_{n}\right)$ if $n$ is even,
$\mathrm{F}_{i}=\mathrm{F}\left(x_{1}^{i},x_{2}^{i},\ldots,x_{n}^{i}\right)-\left(\frac{a_{p}}{2}x_{p}+a_{p+1}x_{p+1}+\ldots+a_{n}x_{n}\right)$ if $n$ is odd.
We then see that the expression whose characteristic polynomial is

\left.\left.\left|\begin{array}[]{llll}\mathrm{F}_{1}&\mathrm{\penalty 10000\ F}_{2}&\ldots&\mathrm{\penalty 10000\ F}_{p+1}\\ \mathrm{\penalty 10000\ F}_{2}&\mathrm{\penalty 10000\ F}_{3}&\ldots&\mathrm{\penalty 10000\ F}_{p+2}\\ \ldots&\ldots&\ldots&\ldots\\ \mathrm{\penalty 10000\ F}_{p+1}&\mathrm{\penalty 10000\ F}_{p+2}&\ldots&\mathrm{\penalty 10000\ F}_{2p+1}\end{array}\right|=\mathrm{C}x_{1}x_{2}\ldots x_{p}\right\rvert\,\mathrm{V}\left(1,x_{1},x_{2},\ldots x_{p}\right)\right]^{2}

is a consequent of (1). Here C is a (non-zero) constant equal to $a_{0}a_{1}\ldots a_{p}$ Or $a_{0}a_{1}\ldots a_{p-1}\frac{a_{p}}{2}$ , depending on whether $n$ is even or odd and $V\left(\theta_{1},\theta_{2},\ldots,\theta_{k}\right)$ is the Vandermonde determinant of numbers $\theta_{1},\theta_{2},\ldots\theta_{k}$ . The expression whose characteristic polynomial is $\left[V\left(1,x_{1},x_{2},\ldots,x_{p}\right)\right]^{2}$ is equivalent to the previous one, so is still a consequent of (1). This last expression is indeed of the form (5). Of course, the $\%$ of these two corresponding expressions (1) and (5) are not the same. The a of the expression obtained are $\alpha_{1},\alpha_{2},\ldots,\alpha_{p}$ And $\alpha_{i}-\alpha_{j},j=1,2,\ldots,i-1,i=1$ , $2,\ldots,p$ , each taken twice.
4. - We will say that expression (1) is reducible if we can find a consequent of the form (2) (or a consequent equivalent to (2)).

Expression (1) is, in general, reducible. Let always $\mathrm{F}\left(x_{1},x_{2},\ldots,x_{n}\right)$ the characteristic polynomial (of the second type) of (1). In general, we can find $n$ polynomials $A_{i}\left(x_{1},x_{2},\ldots,x_{n}\right)$ , $i=1,2,\ldots,n$ such that we have

\sum_{i=1}^{n}\mathrm{\penalty 10000\ A}_{i}\left(x_{1},x_{2},\ldots,x_{n}\right)\mathrm{F}\left(x_{1}^{i},x_{2}^{i},\ldots,x_{n}^{i}\right)=\Phi\left(x_{1}\right)

$\Phi\left(x_{1}\right)$ being a polynomial ell $x_{1}$ alone. This means that there exists a consequent of the form

\Sigma a_{i}^{\prime}f\left(x+i\alpha_{1}h\right)

(7)

which is also of the form (4) and has the characteristic polynomial
of the first type $\Phi\left(x_{1}\right)$ . It is obvious that an expression is reducible if it has a reducible consequent. In the next No. we will demonstrate that expressions (4) are reducible, our stated property is therefore demonstrated.

We know from elimination theory that we can find the polynomials $A_{i}$ such as $\Phi\left(x_{1}\right)$ either of degree $n!$ . The order of expression (7) is 0 in general, but if expression (1) is of order $>0$ , (7) ext at least of order $n$ , which follows immediately from the fact that the coefficients $a_{i}^{\prime}$ do not depend on numbers $\alpha_{i}$ . We can also easily see that if expression (1) is of order $>0$ , the polynomial $\Phi\left(x_{1}\right)$ is of the form $\mathrm{C}(x-1)^{n!}$ , C being a constant, reducibility is therefore demonstrated. In special cases the elimination can give a polynomial $\Phi\left(x_{1}\right)$ of degree $<n$ !, still in good shape $\mathrm{C}(x-1)^{\mu}$ if (1) is of order $>0$ , but this polynomial is at least of degree $n$ , unless it is identically zero. These remarks do not apply to expressions (1) of order 0.

There are exceptional cases where the previous reasoning no longer applies. It may, in fact, happen that the polynomial $\boldsymbol{\Phi}\left(x_{1}\right)$ is zero identically. In this case we are tempted to look, first of all, for other values for the positive integers $p_{1},p_{2},\ldots,p_{n}$ such that we have a relation of the form

\text{ (8) }\quad\sum_{i=1}^{n}\mathrm{\penalty 10000\ A}_{i}\left(x_{1},x_{2},\ldots,x_{n}\right)\mathrm{F}\left(x_{1}^{p},x_{2}^{p},\ldots,x_{n}^{p_{i}}\right)=\text{ polynome }

but it follows from what follows that this equality is not possible if $\Phi\left(x_{1}\right)$ is zero identically.

We can clearly recognize whether we are in this exceptional case by the following property:

The necessary and sufficient condition for the polynomial $\Phi\left(x_{1}\right)$ is zero identically is that we can find two equalities of the form

	$\displaystyle a_{0}+a_{\mu_{1}}+a_{\mu_{2}}+\cdots+a_{\mu_{i}}=0$		(9)
	$\displaystyle a_{1}+a_{\nu_{1}}+a_{\nu_{2}}+\cdots+a_{\nu_{j}}=0$		( $\prime$ )

Or $i\geqq 1,j\geqq 1$ and the $\mu_{1},\mu_{2},\ldots,\mu_{i},\nu_{1},\nu_{2},\ldots,\nu_{j}$ are all distinct and chosen from the numbers 2, 3, . . . , n.

It is easy to see that the condition is sufficient. Let us show that it is also necessary. We will demonstrate this property
by complete induction. The fact that the polynomial $\left.\Phi_{(}^{\prime}x_{1}\right)$ is zero identically means that the system

F\left(x_{1}^{i},x_{2}^{i},\ldots,x_{n}^{i}\right)=0,\quad i=1,2,\ldots,n

(10)

has a solution in $x_{2},x_{3},\ldots,x_{n}$ for any value of $x_{1}$ . For this to be so, it is obviously sufficient that this property be true for an infinity of values of $x_{1}$ . Let us also note that there then certainly exists a finite solution in $x_{2},x_{3},\ldots,x_{n}$ for everything $x_{1}$ , according to the very form of equations (10). Let us also recall that we always assume $\alpha_{i}\neq 0.i=0,1,\ldots,n$ .

The property is true for $n=2$ (and also for $n=1$ ) since in this case $\Phi\left(x_{1}\right)$ cannot be identically zero. Suppose the property is true until $n-1$ and let's demonstrate it for $n$ . The (algebraic) system (10) shows us that there certainly exists an interval $(c,d)$ and functions $x_{2}=x_{2}\left(x_{1}\right),x_{3}=x_{3}\left(x_{1}\right),\ldots$ ..,, $x_{n}=x_{n}\left(x_{1}\right)$ of $x_{1}$ , continuous and differentiable when $x_{1}$ is in ( $c,d$ ) and which checks the system for $c<x<d$ . Substituting these values into the system (10) and deriving with respect to $x_{4}$ , we find

\sum_{i=1}^{n}a_{i}x_{i}^{j}x_{i}^{\prime}=0,\quad j=0,1,\ldots,n-1\quad\left(x_{1}^{\prime}=1\right)

This system must be compatible in $x^{\prime}{}_{2},x^{\prime}{}_{3},\ldots,x^{\prime}{}_{n}$ and we deduce that $V\left(x_{1},x_{2},\ldots,x_{n}\right)=0$ For $c<x_{1}<d$ . It is therefore necessary that, for an infinity of values of $x_{1}$ , at least two of the variables $x_{4},x_{2},\ldots,x_{n}$ are equal. We see that our problem is thus reduced to the same problem where $n$ is smaller. If we get to the case $n=2$ , SO $\mathrm{F}\left(x_{1},x_{2}\right)=a_{0}+a_{1}^{\prime}x_{1}+a_{2}^{\prime}x_{2}$ it is necessary that $a_{1}^{\prime}=0$ , $a_{0}+a_{2}^{\prime}=0$ which are exactly conditions (9) and (9'). It should be noted, however, that if $x_{1}$ is equal to one of the variables $x_{2},x_{3},\ldots,x_{n}$ an infinite number of times, it may be that the new system (10) is always of the form (10) where however $\alpha_{1}=0$ . We then see that equality (9') is already demonstrated and it remains to establish equality (9). Now, this equality can be obtained very simply, and independently of the previous considerations, by noting that the system

F\left(0,x_{2}^{i},x_{3}^{i},\ldots,x_{n}^{i}\right)=0,\quad i=1,2,\ldots,n

tion, must be equal to 1. We then proceed by induction. The property is completely demonstrated.

If $\Phi\left(x_{1}\right)$ is zero identically we can look for the eliminating in one of the other variables $x_{2},x_{j},\ldots,x_{n}$ . Moreover, we can take as the first coefficient $a_{0}$ any of the other coefficients $a_{i}$ . It may, of course, happen that all the eliminators thus obtained are identically zero. Finally, in certain particular cases, we can demonstrate the reducibility of expression (1) by looking for an eliminator in $2,3,\ldots,n-1$ variables. We do not insist on these cases.

For $n=3$ expression (1) is always reducible, unless the coefficients are proportional to the numbers $1,1,-1,-1$ . For $n=4$ the expression is reducible, unless its coefficients are proportional to the numbers 2, 1, -1, -1, -1. For $n=5$ the problem is already more complicated. Any expression 1) is reducible in this case, unless the coefficients are proportional to the numbers in one of the following groups

1,	1,	1,	-1,	-1,
3,	1,	-1,	-1,	-1,
1,	-1,	$\lambda$ ,	$-\lambda$ ,	$1:\lambda$ ,
1,	-1,	$\lambda$ ,	$\lambda$ ,	$1-\lambda$ ,
1,	1,	$\lambda$ ,	$\lambda$ ,	$-1-\lambda$
1,			$-1-\lambda$	$-1-\lambda$

$\lambda$ being any number. We verify that, for $n=3,4,5$ , any expression of order $n$ is reducible. Therefore, for $n=3,4,5$ , we know that expression (3) is reducible. This is most likely the case for $n$ any. We believe, moreover, that it exists, for everything $n$ , a number $\mathrm{N}(n)<n$ such that any expression (1) of order $>N(n)$ is surely reducible. The determination of this number $\mathrm{N}(n)$ is an algebraic problem whose resolution appears to present certain difficulties.
5. - It remains to demonstrate that expression (4) is reducible. We have already demonstrated this property in our previous work [11]. We will specify our results here.

Let us first prove the following lemma:
If $\mathrm{F}^{*}(1)\neq 0$ , there are an infinity of positive integers $p$ such as $\mathrm{F}^{*}(x),\mathrm{F}^{*}\left(x^{p}\right)$ be first among themselves.

Here $\mathrm{F}^{*}(x)$ is the characteristic polynomial of the first type of expression (4). This polynomial is of the form $\mathrm{F}^{*}(x)=\prod_{i=1}^{n^{\prime}}\left(x-\sigma_{i}\right)^{\mu_{i}}$ , Or $\sigma_{1},\sigma_{2},\ldots,\sigma_{n^{\prime}}$ are $n^{\prime}$ distinct numbers (real or complex) and different from 1 and $\sum_{i=1}^{n^{\prime}}\mu_{i}=n$ . Let us first assume $\left|\sigma_{i}\right|=1,i=1$ , $2,\ldots,n^{\prime}$ and be $\theta_{1},\theta_{2},\ldots,\theta_{n}$ the arguments, between 0 and $2\pi\left(0<\theta_{i}<2\pi\right)$ , numbers $o_{1},o_{2},\ldots,o_{n^{\prime}}$ . Let us assume, in general, that $\sigma_{1},\sigma_{2},\ldots,\sigma_{j}$ are primitive roots of the unity of orders $q_{1},q_{2},\ldots,q_{j}$ respectively and that the others $o_{i}$ are not roots of unity. If $o_{r}$ And $o_{s}$ are not both roots of unity one cannot have $p\theta_{r}\equiv\theta_{s}(\bmod 2\pi)$ that at most for a value of $p(>1)$ . If $o_{r}$ , os are both roots of unity we have $p9_{r}\equiv\theta_{s}(\bmod 2\pi)$ if $p$ is a multiple of $q_{1}q_{2}\ldots q_{j}$ . The lemma follows in this case. It remains to examine the case where the $\sigma_{i}$ are any $\neq 1$ . This case results from the previous one since if $\left|\sigma_{r}\right|,\left|\sigma_{s}\right|$ are not both equal to 1, we surely have $o_{r}^{p}\neq\sigma_{s}$ For $p$ large enough.

Now consider the case of any order $k$ . We can write $F^{*}(x)=(x-1)^{k}F_{1}^{*}(x)$ , Or $F_{1}^{*}(1)\neq 0$ . We can therefore find an integer $p$ such as $\mathrm{F}_{1}^{*}(x),\mathrm{F}_{1}^{*}\left(x^{p}\right)$ are coprime, therefore also two polynomials $\phi(x),\psi(x)$ such as

\phi(x)\mathrm{F}_{1}^{*}(x)+\psi(x)\mathrm{F}_{1}^{*}\left(x^{p}\right)\equiv 1.

We immediately deduce that there are two polynomials $\phi_{1}(x),\psi_{1}(x)$ such as

\phi_{1}(x)\mathrm{F}^{*}(x)+\psi_{1}(x)\mathrm{F}^{*}\left(x^{p}\right)\equiv\left(x^{p}-1\right)^{k}

and we can state the following property:
Any expression (4), of order $k$ , has a consequent of the form $\Delta_{h}^{k}f(x)$ , of order $k$ and of the form (2).

The previous reduction to the form (5) shows us that if among the numbers $\alpha_{i}$ there is at least $\left|\frac{n+1}{2}\right|$ which have their mutual rational relations, expression (1) is reducible. This is always the case if $n=2$ , so any expression 1) with three terms is reducible.
6. - We can extend the previous results to the case of several variables. Given a function $f\left(x_{1},x_{2},\ldots,x_{m}\right)$
of $m$ variables $x_{1},x_{2},\ldots,x_{m}$ , the operation $\Delta_{h_{1},h_{2},\ldots,h_{m}}^{\left(a_{1},a_{2i},\ldots,a_{mi}\right)}$ has the following meaning

\Delta_{h_{1},h_{2},\ldots,h_{m}}^{\left(\alpha_{1i},\alpha_{2i},\ldots,\alpha_{mi}\right)}f\left(x_{1},x_{2},\ldots,x_{m}\right)=\Sigma^{*}a_{i_{1}i_{2}\ldots i_{m}}f\left(\ldots,x_{j}+\alpha_{ji_{j}}h_{j},\ldots\right)

(11)

where the summons $\Sigma^{*}$ is extended to values $i_{1}=0,1,\ldots,n_{1}$ , $i_{2}=0,1,\ldots,n_{2},\ldots,i_{m}=0,1,\ldots,n_{m}$ . To shorten the writing, we will often ask $f\left(\ldots,\xi_{j},\ldots\right)$ instead of $f\left(\xi_{1},\xi_{2},\ldots,\xi_{m}\right)$ . Such an operation is characterized by the (multiple) sequence of (real) coefficients $a_{i_{1}i_{2}\ldots i_{m}}$ and by $m$ sequences of pseudo-periods (real) $\alpha_{j0},\alpha_{j1},\ldots,\alpha_{jn_{j}},j=1,2,\ldots,m$ . Each of these suites presents a character of homogeneity. The $x_{1},x_{2},\ldots,r_{m}$ and the $h_{1},h_{2},\ldots,h_{m}$ are variables, which specifies the equivalence of two operations. It is still advantageous to consider an operation as having meaning only if the coefficients are not all zero. We then have the property:

The product of two operations always has a meaning, is still an operation of the same nature and this multiplication is commutative.

This property also applies not only to operations that operate on all variables but also to those that operate on some of these variables.

When we consider the general expression (11) we can assume that $\alpha_{10}=\alpha_{20}=\ldots=\alpha_{m0}$ and that $\alpha_{jr}\neq\alpha_{js},r\neq s,j=1$ , $2,\ldots,m$ . As for the coefficients, we can assume that we have

	$\displaystyle\sum_{i_{1}=0}^{n_{1}}\cdot\sum_{i_{j-1}=0}^{n_{j-1}}\sum_{i_{j+1}=0}^{n_{j+1}}\sum_{i_{m}=0}^{n_{m}}\left\|a_{i_{1}\ldots i_{j-1}i_{j}i_{j+1}\ldots i_{m}}\right\|\neq 0,$		(12)
	$\displaystyle i_{j}=0,1,\ldots,n_{j},\quad j=1,2,\ldots,m.$

Let's look at some special cases. The expression

		$\displaystyle\Delta_{h_{1}}^{n_{1}},n_{2},\ldots,h_{m},\ldots,h_{m}f\left(x_{1},x_{2},\ldots,x_{m}\right)=$		(13)
	$\displaystyle=$	$\displaystyle\Sigma^{*}(-1)^{n_{1}+n_{2}+\cdots+n_{m}-i_{1}-i_{2}-\cdots-i_{m}}\binom{n_{1}}{i_{1}}\binom{n_{2}}{i_{2}}\ldots\binom{n_{m}}{i_{m}}f\left(\ldots,x_{j}+i_{j}h_{j},\ldots\right)$

is a difference of order ( $n_{1},n_{2},\ldots,n_{m}$ ) of the function $f\left(\ldots,x_{j},\ldots\right)$ . We obtain a more general case by considering operations of the form $\Delta_{h_{1}}^{\left(a_{1i}\right)}\Delta_{h_{2}}^{\left(\alpha_{2i}\right)}\ldots\Delta_{h_{m}}^{\left(\alpha_{mi}\right)}$ , Or $\Delta_{h_{j}}^{\left(a_{ji}\right)}$ operates on the variable $x_{j}$ .

Let's ask $\phi_{j}(x)=\left(x-\alpha_{j0}\right)\left(x-x_{j1}\right)\ldots\left(x-x_{jn}\right)$ , we have the expression

\Sigma^{*}\frac{f\left(\ldots,x_{j}+\alpha_{ji_{j}}h_{j},\ldots\right)}{\phi_{1}^{\prime}\left(\alpha_{1i_{1}}\right)\phi_{2}^{\prime}\left(\alpha_{2i_{2}}\right)\ldots\phi_{m}^{\prime}\left(\alpha_{mi_{m}}\right)}

(14)

which is of this form. Another expression of this form, which we will use later, is

	$\displaystyle\delta_{h}^{\left(\alpha_{1i},\alpha_{2i},\ldots,\alpha_{mi}\right)}f\left(x_{1},x_{2},\ldots,x_{m}\right)=$		(15)
	$\displaystyle\quad=\delta_{h_{1}}^{\left(\alpha_{1i}\right)}\delta_{h_{2}}^{\left(\alpha_{2i}\right)}\ldots\delta_{h_{m}}^{\left(\alpha_{mi}\right)}f\left(x_{1},x_{2},\ldots,x_{m}\right)$

Or $\delta_{h_{j}}^{\left(a_{ji}\right)}$ is operation (5), operating on the variable $x_{j}\left(h=h_{j},n=n_{j}\right)$ . Finally, if $\alpha_{ji}=i,i=0,1,\ldots,n_{j},j=1,2,\ldots,m$ , we have the expression

\Delta_{h_{1},h_{2},\ldots,h_{m}}^{\left(n_{1},n_{2},\ldots,n_{m}\right)}f\left(x_{1},x_{2},\ldots,x_{m}\right)

(16)

where it is now unnecessary to make the restrictions (12).
7. - Let us now define and specify the order of an expression (11). This order is $\left(k_{1},k_{2},\ldots,k_{m}\right)$ Or $k_{j}$ is the minimum of the orders of the expressions

\begin{gathered}\sum_{i_{j}=0}^{n_{j}}a_{i_{1}\ldots i_{j}\ldots i_{m}}\phi\left(x+\alpha_{i_{j}}h\right)\\ 0\leqq i_{1}\leqq n_{1},\ldots,0\leqq i_{j-1}\leqq n_{j-1},0\leqq i_{j+1}\leqq n_{j+1},\ldots,0\leqq i_{m}\leqq n_{m}.\end{gathered}

In this way, each variable $x_{j}$ contributes by a number $k_{j}$ to the definition of order. We will also say that the expression has simple order $\left[k_{j}-1\right]_{j}$ compared to $x_{j}$ .

Let's separate the variables $x_{1},x_{2},\ldots,x_{m}$ in two groups $x_{j_{1}},x_{j_{2}},\ldots,x_{j_{r}};x_{j_{r+1}},\ldots,x_{j_{m}},j_{1}<j_{2}<\ldots<j_{r};j_{r+1}<j_{r+2}<\ldots<j_{m}$ .
We will use similar separations in the following. To simplify the writing, let us put $n_{s}^{\prime}$ For $n_{j_{s}}$ And $i_{s}^{\prime}$ For $i_{j_{s}}$ . Let us then introduce the following notations ( ¹ )
$\left.\gamma_{i_{1}^{\prime}i_{2}^{\prime}\ldots i_{r}^{\prime}}^{\left(v_{r+1},v_{r+2}\right.},\ldots,v_{m}\right)=\sum_{i^{\prime}{}_{r+1}=0}^{n_{r+1}^{\prime}}\sum_{i^{\prime}{}_{r+2}=0}^{n^{\prime}r+2}\sum_{i^{\prime}m=0}^{n^{\prime}m}a_{i_{1}i_{m}\ldots i_{m}}\alpha_{j_{r+1}i_{r+1}^{\prime}}^{v_{r+1}}\alpha_{j_{r+2}i_{r+2}}^{v_{r+2}}\ldots\alpha_{j_{m}i_{m}}^{v_{m}}$
(') For $r=0$ we have the numbers $\gamma^{\left(v_{1},v_{2},\ldots,v_{m}\right)\text{. }}$

The number $k_{1}$ of the order is then characterized by the relations

\begin{gathered}\gamma_{i_{2}i_{3}\ldots i_{m}}^{(\nu)}=0,\nu=0,1,\ldots,k_{1}-1,0\leqq i_{j}\leqq n_{j},j=2,3,\ldots,m\\ \sum_{i_{2}=0}^{n_{2}}\sum_{i_{3}=0}^{n_{3}}\ldots\sum_{i_{m}=0}^{n_{m}}\left|\gamma_{i_{2}i_{3}\ldots i_{m}}^{\left(k_{1}\right)}\right|\neq 0.\end{gathered}

These relationships therefore define the simple order $\left[k_{1}-1\right]_{1}$ . We will now introduce other orders, double, triple, . . . multiple which characterize the expression (11).

We will say that $\left[k_{1}^{\prime},k_{2}^{\prime}\right]_{1,2}$ is a double order (relative to the variables $x_{1},x_{2}$ ) of expression (11) if $k_{1}^{\prime}\geq k_{1},k_{2}^{\prime}\geq k_{2}$ and if
(17) $\gamma_{i_{3}i_{4}\ldots i_{m}}^{\left(v_{1},v_{2}\right)}=0,v_{1}=0,1,\ldots,k_{1}^{\prime},v_{2}=0,1,\ldots,k_{2}^{\prime},0\leqq i_{j}\leqq n_{j},j=3,4,\ldots,m$
$\sum_{i_{3}=0}^{n_{3}}\sum_{i_{4}=0}^{n_{4}}\ldots\sum_{i_{m}=0}^{n_{m}}\left|\gamma_{i_{3}}^{\left(k^{\prime}i_{4}+1,k^{\prime}{}_{2}\right)}\right|\neq 0,\sum_{i_{5}=0}^{n_{8}}\sum_{i_{4}=0}^{n_{4}}\ldots\sum_{i_{m}=0}^{n_{m}}\left|\gamma_{i_{3}i_{4},\ldots i_{m}}^{\left(k_{1}^{\prime},k^{\prime},i_{m}\right)}\right|\neq 0$ .
It is useless to consider the case $k_{1}^{\prime}<k_{1}$ or the case $k_{2}^{\prime}<k_{2}$ , since

\gamma_{i_{3},\ldots i_{4}}^{\left(\nu_{1},\nu_{2}\right)}=\sum_{i_{2}=0}^{n_{3}}\gamma_{i_{2}i_{3}\ldots i_{m}}^{\left(\nu_{1}\right)}\alpha_{2i_{2}}^{\nu_{2}}=\sum_{i_{1}=0}^{n_{1}}\gamma_{i,i_{3}\ldots i_{m}}^{\left(\nu_{2}\right)}\alpha_{1i_{1}}^{\nu_{1}}

and equality (17) is then a consequence of the definition of order.

We determine the double orders in the following way. Let $k_{1}^{\prime}\left(\geqq k_{1}\right)$ given and consider the expressions

\sum_{i_{2}=0}^{n_{2}}\sum_{i_{3}=0}^{n_{9}}\ldots\sum_{i_{m}=0}^{n_{m}}\gamma_{i_{2}i_{3}\ldots i_{m}}^{\left(v_{1}\right)}f\left(\xi_{2},\xi_{3},\ldots,\xi_{m}\right),\quad y_{1}=k_{1},k_{1}+1,\ldots,k_{1}^{\prime},

by agreeing to pose, here and hereafter, $\xi_{s}=x_{s}+\alpha_{si}h_{s}$ . Each of these expressions has a simple order, namely $\left[s^{\left(p_{1}\right)}\right]_{2}$ , compared to $x_{2}$ . The minimum number $s^{\left(v_{1}\right)}$ is the number $k_{2}^{\prime}$ . We see, in fact, that the definition of the number $k_{2}^{\prime}$ is identical to the definition (17. We also deduce the following property:

The necessary and sufficient condition for expression
(11), of order ( $k_{1},k_{2},\ldots,k_{m}$ ), does not have double orders is that the expressions
(18) $\sum_{i_{1}=0}^{n_{1}}\ldots\sum_{i_{j-1}=0}^{n_{j-1}}\sum_{i_{j+1}=0}^{n_{j+1}}\sum_{i_{m}=0}^{n_{m}}\gamma_{i_{1}\ldots i_{j-1}i_{j+1}\ldots i_{m}}^{\left(k_{j}\right)}f\left(\xi_{1},\ldots,\xi_{j-1},\xi_{j+1},\ldots,\xi_{m}\right)$
be of order $\left(k_{1},\ldots,k_{j-1},k_{j+1},\ldots,k_{m}\right),j=1,2,\ldots,m$ .

Conversely, if expressions (18) are of order ( $k_{1},\ldots,k_{j-1}$ ; $k_{j+1},\ldots,k_{m}$ ), expression (11) is of order ( $k_{1}^{*},k_{2}^{*},\ldots,k_{m}^{*}$ ) Or $k_{j}^{*}\leqq k_{j}$ .

We can define, in general, the ruple orders of the expression (11). The symbol $\left[k_{1}^{\prime},k_{2}^{\prime},\ldots,k_{r}^{\prime}\right]_{1,2,\ldots,r}$ represents a ruple order (relative to the variables $x_{1},x_{2},\ldots,x_{r}$ ) if
$1^{n}$ . None of the symbols $\left[k_{1}^{\prime},\ldots,k_{s-1}^{\prime},k_{s+1}^{\prime},\ldots,k_{r}^{\prime}\right]_{1,\ldots,s-1,s+1,\ldots,r}$ , $s=1,2,\ldots,r$ is not an order ( $r-1$ ; uple and, more generally, none of the symbols $\left[k_{i_{1}}^{\prime},k_{i_{2}}^{\prime},\ldots,k_{i_{s}}^{\prime}\right]_{i_{1},i_{2},\ldots,i_{s}}$ is not an order $s^{\text{upple }}$ . In particular done $k_{s}^{\prime}\geqq k_{s}$ .
$2^{0}$ . We have

	$\displaystyle\gamma_{i_{r+1}i_{r+2}\ldots i_{n}}^{\left(v_{1},v_{2},\ldots,v_{r}\right)}=0,v_{s}=0,1,\ldots,k_{s}^{\prime},s=1,2,\ldots,r$		(19)
	$\displaystyle 0\leqq i_{j}\leqq n_{j},j=r+1,r+2,\ldots,m$
	$\displaystyle\left.\sum_{i_{r+1}=0}^{n_{r+1}}\sum_{i_{r+2}=0}^{n_{r+2}}\sum_{i_{m}=0}^{n_{m}}\mid\gamma_{i_{r+1},\ldots,k_{s}^{\prime}+1,\ldots,k_{r}^{\prime}}^{n_{r}}\right)\mid\neq 0,s=1,2,\ldots,r.$

A part of the equalities (19) is moreover a consequence of the existence of simple, double orders, $\ldots.,(r-1)^{\text{uples }}$ .

We can still determine the triple orders very simply. Let us suppose that $\left[k_{1}^{\prime},k_{2}^{\prime},\ldots,k_{r-1}^{\prime}\right]_{1,2,\ldots,r-1}$ not be an order $(r-1)^{\text{uple. }}$ . Let us then consider the expressions

\sum_{i_{r}=0}^{n_{r}}\sum_{i_{r+1}=0}^{n_{r+1}}\sum_{i_{m}=0}^{n_{m}}\gamma_{i_{r}i_{r+1}\ldots i_{m}}^{\left(v_{1},v_{2},\ldots,v_{r-1}\right)}f\left(\xi_{r},\xi_{r+1},\ldots,\xi_{m}\right)

(20)

$v_{1}=k_{1},k_{1}+1,\ldots,k_{1}^{\prime},v_{2}=k_{2},k_{2}+1,\ldots,k_{2}^{\prime},\ldots,v_{r-1}=k_{r-1},k_{r-1}+1,\ldots,k_{r-1}^{\prime}$ .
Each has a simple order relative to $x_{r}$ . The minimum of orders is the number $k^{\prime}{}_{r}$ . Of course, we only consider expressions (20) which do not all have zero coefficients. It is possible, in fact, due to the existence of double orders, $\ldots,(r-1)^{\text{uples }}$ , that some of the expressions (20) do not make sense.

We can also see, step by step, that:
The necessary and sufficient condition for the expression (11), of order ( $k_{1},k_{4},\ldots,k_{m}$ ), does not have double, triple, …, ruple orders is that the expressions

\sum_{i^{\prime}=0}^{n_{r}^{\prime}}\sum_{i_{r}^{\prime}+1=0}^{n_{r+1}^{\prime}\ldots}\sum_{i^{\prime}m=0}^{n^{\prime}m}\gamma_{i_{r}^{\prime}i_{r}^{\prime}+1\ldots i_{ni}^{\prime}}^{\left(k_{j_{1}},k_{j_{1}},\ldots,k_{r-1}\right)}f\left(\xi_{j_{r}},\xi_{j_{r+1}},\ldots,\xi_{j_{m}}\right)

(21)

are all of orders $\left(k_{j},k_{j},\ldots,k_{j_{r-1}}\right)$ .

We call the expression

\sum_{i^{\prime}r+1=0}^{n_{r+1}^{\prime}}\sum_{i^{\prime}r+2=0}^{n_{r+2}^{\prime}}\sum_{i^{\prime}m_{m}=0}^{n_{m}^{\prime}m}\gamma_{i_{r+1}^{\prime}}^{(0,0,\ldots,0)}f\left(i_{r+2}^{\prime}\ldots i^{\prime}mf\left(\xi_{r+1},\ldots,\xi_{j_{m}}\right)\right.

(22)

tine rème expression derived from (11). This expression exists only if $k_{j_{1}}=0,k_{j_{1}}=0,\ldots,k_{j_{r}}=0$ .

We have the following property, which we will use later:

If all the derived expressions are of order ( $0,0,\ldots,0$ ), expression (11) itself and all the $s^{\text{èmes }}$ derived expressions, with $s=1,2,\ldots,r-1$ , are also of order ( $0,0,\ldots,0$ ).
8. - The notion of characteristic polynomial can be extended to the case of $m$ variables. We have the characteristic polynomial of the first type

\mathrm{F}^{*}\left(x_{1},x_{2},\ldots,x_{m}\right)=\Sigma^{*}a_{i_{1}i_{2}\ldots i_{m}}x_{1}^{a_{1i_{1}}}x_{2}^{a_{2i_{1}}\ldots x_{m}^{a_{mi_{m}}}}

which in the case of expression (16) is a polynomial properly speaking. The order ( $k_{1},k_{2},\ldots,k_{m}$ ) is then characterized by the relations

\begin{gathered}{\left[\frac{\partial^{v_{j}}\mathrm{\penalty 10000\ F}^{*}}{\partial x_{j}^{v_{j}}}\right]_{x_{j}=1}\equiv 0,\quad v_{j}=0,1,\ldots,k_{j}-1,\quad\left[\frac{\partial^{k_{j}}\mathrm{\penalty 10000\ F}^{*}}{\partial x_{j}^{k_{j}}}\right]_{x_{j}=1}\equiv_{i}^{\prime}\equiv 0}\\ j=1,2,\ldots,m,\end{gathered}

the identity or non-identity always being with respect to the remaining variables.

For example, in the case of expression (15), we have

\left.\mathrm{F}^{*}\left(x_{1},x_{2},\ldots,x_{m}\right)-\prod_{i=1}^{m}\prod_{i=1}^{n_{j}}1-x_{i}^{a_{ji}}\right)

and this expression is, therefore, of order ( $n_{1},n_{2},\ldots,n^{\prime\prime}$ ).
The characteristic polynomial of the derived expression (22) is obtained from $F^{*}\left(x_{1},x_{2},\ldots,x_{m}\right)$ by doing it $x_{j_{1}}=x_{j_{2}}=\ldots=x_{j_{r}}=1$ .

The characteristic polynomial of expression (20) is obtained by doing $x_{1}=x_{2}=\cdots=x_{r-1}=1$ In

\left(x_{1}\frac{\partial}{\partial x_{1}}\right)^{\left(v_{1}\right)}\left(x_{2}\frac{\partial}{\partial x_{2}}\right)^{\left(v_{2}\right)}\cdots\left(x_{r-1}\frac{\partial}{\partial x_{r-1}}\right)^{\left(v_{r-1}\right)}\mathrm{F}^{*}\left(x_{1},x_{2},\ldots,x_{m}\right)

We easily deduce that for the existence of a ruplicated order, $\left[k_{1}^{\prime};k_{2}^{\prime};\ldots;k_{r}^{\prime}\right]_{1,2,\ldots,r}$ ; it is necessary that we have

\begin{gathered}{\left[\frac{\partial^{v_{1}}+v_{2}+\cdots+v_{r}\mathrm{\penalty 10000\ F}^{*}}{\partial x_{1}^{v_{1}}\partial x_{2}^{v_{2}}\cdot\cdot\partial\cdot x_{r}^{v_{r}}}\right]_{x_{1}=x_{2}=\cdots=x_{r}=1}\equiv 0}\\ v_{s}=0,1,\ldots,k_{s}^{\prime},\quad s=1,2,\ldots,r\end{gathered}

Consider again expression (15). The characteristic polynomial of expression (22) is then (up to a constant factor)

$\coprod_{s=r}^{m}\prod_{i=1}^{n_{j}}\left(1-x_{j_{s}}^{\alpha_{j}s_{i}}\right)$ and we see that

Expression (15) has no double, triple, … multiple orders:
We can easily deduce that in a rupto order, $\left[k_{1}^{\prime},k_{2}^{\prime},\ldots,k_{r}^{\prime}\right]_{1,2,\ldots,r}$ , we must have $k_{s}^{\prime}<n_{s}$ .
9. - Let us also call expression associated with (11) any expression of the form

\Sigma bf\left(\ldots;x_{j}+\beta_{ii}h_{j},\ldots\right)

(23)

where the $b$ are the coefficients and the $\beta_{ji}$ are of the form $r_{1}^{(j)}x_{j1}+r_{2}^{(j)}x_{j2}+\ldots+r_{n_{j}}^{(j)}x_{n_{j}}$ , THE $r_{s}^{(j)}$ being positive or zero integers. We will then say that the polynomial, at $n_{1}+n_{2}+\ldots+n_{m}$ variables,

\Sigma bx_{11}^{r_{1}^{(1)}}x_{12}^{r_{2}^{(1)}}\ldots x_{1n_{1}}^{r_{1}^{(1)}}x_{21}^{r_{1}^{(2)}}\ldots x_{2n_{2}}^{r_{2}^{(2)}}\ldots x_{m_{1}}^{r_{1}^{(m)}}\ldots x_{mn_{m}}^{r_{m}^{(m)}}

is the characteristic polynomial of the second type of expression (23).
In particular, expression (11) itself has the characteristic polynomial of the second type

\Sigma^{*}a_{i_{1}i_{1}\ldots i_{m}}x_{1i_{1}}x_{2i_{2}}\ldots x_{mi_{m}}\left(x_{10}=x_{20}=\ldots=x_{m0}=1\right)

The characteristic polynomial of the associated expression (15) is $\prod_{i=1}^{m}\prod_{i=1}^{n_{j}}\left(1-x_{ii}\right)$ .

We still define, as in the case of a single variable, the consequent expressions using the characteristic polynomials (of the first and second type).

Let us now demonstrate the generalization of the property of No. 3:

Every expression (11) has a consequent of the form (15).

To avoid unnecessary complication, it will be sufficient to give the demonstration in the case of two variables ( $m=2$ ). Let us then pose $n_{1}=m,n_{2}=n$ ; the characteristic polynomial can be written

\mathrm{F}\left(x_{11},x_{12},\ldots,x_{1n_{1}}\mid x_{21},x_{22},\ldots,x_{2n_{2}}\right)=\sum_{i=0}^{m}\left(\sum_{j=0}^{n}a_{ij}x_{2j}\right)x_{1i}

In this form this polynomial is of the form $\mathrm{F}\left(x_{1},x_{2},\ldots,x_{n}\right)$ from No. 2. The coefficients $\sum_{j=0}^{n}a_{ij}x_{2j},i=0,1,\ldots,m$ are not identically zero. Applying the method of No. 3, we see that expression (11) has a consequent having a characteristic polynomial of the form

\left.\left.\mathrm{G}_{\left(x_{21}\right.},x_{22},\ldots,x_{2n}\right)\left[\mathrm{\penalty 10000\ V}_{(1},x_{11},x_{12},\ldots,x_{1p}\right)\right]^{2}

We can in the polynomial $G\left(x_{21},x_{22},\ldots,x_{2n}\right)$ ageimiler each term $x_{21}^{r_{1}}x_{22}^{r_{2}}\ldots x_{2n}^{r_{n}}$ to a variable $x_{2j}$ and we can do the same for $\left[\mathrm{V}\left(1,x_{11},x_{12},\ldots,x_{1p}\right)\right]^{2}$ . In other words, the obtained expression is associated with some expression which is itself associated with (11) and has a characteristic polynomial of the form

\left(\sum_{j=1}^{q}a_{j}^{\prime}x_{2j}\right)\left(1-x_{11}\right)\left(1-x_{12}\right)\ldots\left(1-x_{1p_{1}}\right)\quad\left(x_{20}=1\right)

where we can assume the constants $a_{j}^{\prime}\neq 0$ . Applying once again the method of elimination of No. 3, we find the sought property.

We demonstrate in exactly the same way the property for $m$ any.

Of course, the expression of the form (15) obtained does not have the same a as (11). We can specify the form of this expression, but it is useless to do so here.
10. - We can also say that the expression (11) is reducible if it has a consequent of the form (13) (or a consequent equivalent to (13)). We can look, as in the case of a single variable, for conditions under which an expression (11) is reducible, but we do not insist on this point.

Let us only demonstrate the following property:
Any expression of the form (16) is reducible.

It is still enough to silence the demonstration for $m=2$ . Either $\mathrm{F}^{*}(x,y)=\sum_{i=0}^{m}\sum_{j=0}^{n}a_{ij}x^{i}y^{j}$ the characteristic polynomial of the first type of expression (16). We have, in general, $\mathrm{F}^{*}(x,y)=(1-x)^{k}(1-y)^{k}\mathrm{\penalty 10000\ F}_{1}^{*}(x,y)$ , Or $\mathrm{F}_{1}^{*}x,y$ ) is not divisible by $1-x$ nor by $1-y$ . The order of expression (16) is ( $k,k^{\prime}$ ). The results of No. 5 show us that we can find an integer $p$ and two polynomials $\phi(x,y),\psi(x,y)$ such that we have

\phi(x;y)\mathrm{F}_{1}^{*}(x,y)+\psi(x,y)\mathrm{F}_{1}^{*}\left(x^{p},y\right)=g(y),

g $y$ ) being a (non-identically zero) polynomial in $y$ . Since, by hypothesis, $\mathrm{F}_{1}^{*}(1,y)$ is not identically zero, we conclude the existence of a $p$ for which $\mathrm{F}_{1}^{*}(x,y),\mathrm{F}_{1}^{*}\left(x^{p},y\right)$ are coprime (with respect to $x$ ) for an infinity of values of $y$ (more precisely except perhaps for a finite number of values of $y$ ). This is sufficient for the establishment of the formula. This polynomial $g(y)$ is, in general, of the form $g(y)=y^{l}(1-y)^{l^{\prime}}g(y)$ , $g_{1}(0)\neq 0,g_{1}(1)\neq 0$ . Finally, we see that we can determine two integers $p,q$ and polynomials $\phi_{1}(x,y),\phi_{2}(x,y),\phi_{3}(x,y)$ , $\phi_{4}(x,y)$ such that we have

\begin{array}[]{r}\phi_{1}(x,y)\mathrm{F}^{*}(x,y)+\phi_{2}(x,y)\mathrm{F}^{*}\left(x^{p},y\right)+\phi_{3}(x,y)\mathrm{F}^{*}\left(x,y^{q}\right)+\\ +\phi_{4}(x,y)\mathrm{F}^{*}\left(x^{p},y^{q}\right)=y^{ql}\left(x^{p}-1\right)^{k}\left(y^{q}-1\right)^{k^{\prime}+y^{\prime}}\end{array}

which demonstrates the property.
The demonstration is analogous in the case of any number of variables.
11. - In the definition of expression (11) we assumed that the $h_{1},h_{2},\ldots,h_{m}$ are independent variables. On the contrary, we can assume that the $h_{j}$ are not linearly independent. Suppose that
(24) $h_{j}=\gamma_{j1}h_{1}^{\prime}+\gamma_{j2}h_{2}^{\prime}+\cdots+\gamma_{jk}h_{k}^{\prime},j=1,2,\ldots,m_{3}$
where the numbers $\gamma_{ji}$ are given, the matrix ( $\gamma_{ji}$ ) is of rank $k$ and the $h_{1}^{\prime},h_{2}^{\prime},\ldots,h_{k}^{\prime}$ are independent variables. We then denote the corresponding operation by $\Delta_{h^{\prime}{}_{1},h^{\prime}{}_{2},\ldots,h^{\prime}{}_{k}.}^{\left(a_{1i},a_{2i},\ldots,a_{ni}\right)}$ . We will say that such an operation is $k^{\text{eme }}$ species. In this way the notation $\Delta_{h_{1}}^{\left(\alpha_{1i},h_{2},\ldots,h_{ni},\ldots,\alpha_{ni}\right)}$ means an operation of $m^{\text{ènie }}$ species.
Mathematica, vol. XIV.

If we have relations (24), the first member of (13) will be written $\Delta_{h_{1}^{\prime},h_{1}^{\prime},\ldots,h_{k}^{\prime}}^{n_{1},n_{2},\ldots,n_{m}}f\left(x_{1},x_{2},\ldots,x_{m}\right)$ and we have a difference of order ( $n_{1},n_{2},\ldots,n_{m}$ ) of the kth species.

We will focus our attention particularly on operations or expressions of the first kind, the only ones that we will study in more detail. Such an expression can be written in the form
(25) $\quad\Delta_{h}^{\left(\alpha_{1i},\alpha_{2i},\ldots,\alpha_{mi}\right)}f\left(x_{i},x_{i},\ldots,x_{m}\right)=\sum_{i=0}^{n}a_{i}f\left(\ldots,x_{j}+\alpha_{ji}h,\ldots\right)$ .

Here we can assume $a_{i}\neq 0,i=0,1,\ldots,n,\alpha_{10}=\alpha_{20}=\ldots$ .
$\ldots=\alpha_{m0}=0$ And $\sum_{j=1}^{m}\left|\alpha_{jr}-\alpha\right|\neq 0$ if $r\neq s$ .
An expression associated with (25) will be of the form

\Sigma bf\left(\ldots,x_{j}+\beta_{ji}h,\ldots\right)

(26)

where the $\beta_{ji}$ are of the form $r_{1}\alpha_{j1}+r_{2}\alpha_{j2}+\ldots+r_{n}\alpha_{jn}$ , positive or zero integers $r_{1},r_{2},\ldots,r_{n}$ , being the same for $\beta_{1i},\beta_{2i},\ldots,\beta_{mi}$ .

The characteristic polynomial of (26) is then $\Sigma bx_{1}^{r_{1}}\cdot x_{2}^{r_{3}}\ldots x_{n}^{r_{n}}$ . In particular, the characteristic polynomial of (25) is

\mathrm{F}\left(x_{1},x_{2},\ldots,x_{n}\right)=a_{0}+a_{1}x_{1}+\cdots+a_{n}x_{n}

We see that there is a perfect analogy with the case of a single variable and that the characteristic polynomial plays exactly the same role as the characteristic polynomial of the second type of expressions with one variable. The only difference is that there is a simultaneous correspondence between the terms of the characteristic polynomial and the values of the variables $x_{1},x_{2},\ldots,x_{n}$ . The properties established above therefore apply here with this last precaution. In particular,

Every expression of the first kind (25) has a consequent of the form

\delta_{h}^{\left(a_{1i},a_{i2},\ldots,a_{mi}\right)}f\left(x_{1},x_{2},\ldots,x_{m}\right)=

(27)

=\Sigma(-1)^{i_{1}+i_{2}+\ldots+i_{n}}f\left[\ldots,x_{j}+\left(i_{1}\alpha_{j1}+i_{2}\alpha_{j2}+\cdots+i_{n}\alpha_{jn}\right)h_{,}\ldots\right]

where the summation is extended to all values $i_{r}=0,1$ , $r=1,2,\ldots,n$ .

In expressions of the form (27) it is necessary to assume $\sum_{j=0}^{m}\left|\alpha_{ji}\right|\neq 0$ For $i=1,2,\ldots,n$ .
We will study equations of the first kind in another work. The differences of various kinds will also come into play in the problems treated in chap. V.

CHAPTER II

On a class of functional equations with one variable

12.
- —
  
  We will always assume that these are functions $f(x)$ , real, of the real variable $x$ , uniform and defined in a bounded and open interval $(a,b),a<b$ .

In this chapter we propose to study the functional equation

\Delta_{h}^{\left(\alpha_{i}\right)}f(x)=0

(28)

We will look for the functions $f(x)$ which verify equation (28) for all values of $x$ And $h$ such as $a<x+\alpha_{i}h<b$ , $i=0,1,\ldots,n$ .

In general, we will assume that the numbers $\ddot{\alpha}_{i}$ are distinct, but it is sometimes advantageous not to make this restriction. We always take $\alpha_{0}=0$ . When it comes to the general equation (28) we can assume $a_{i}\neq 0,i=0,1,\ldots,n$ .

We can leave aside the case $n=1$ , when the general solution of the equation is an arbitrary constant or the identically zero function, depending on whether we have $a_{j}+a_{1}=0$ Or $\neq 0$ .

Equation (28) is linear and homogeneous; its solutions therefore enjoy some simple immediate properties such as: the sum of two solutions is still a solution, a solution multiplied by any constant is also a solution etc.

The importance of the notion of order results, first of all, from the following property, which is easily demonstrated:

In the field of polynomials, the general solution of equation (28), of order $k$ , is any polynomial of degree k-1.

A polynomial of degree $k-1$ is an expression of the form $c_{0}x^{k-1}+c_{1}x^{k-2}+\cdots+c_{k-1}$ Or $c_{i}$ are constants, the first $c_{0}$ which can also be zero. For symmetry we will say that the identically zero function is a polynomial of degree -1.

We immediately see that:
Any function verifying equation (28) also verifies any consequent equation.

We deduce, in particular, that:
Any function verifying equation (28) also verifies an equation of the form

\delta_{h}^{\left(a_{i}\right)}f\left(x_{1}=0\right.

(29)

Any function verifying an equation (28) which is reducible, also verifies an equation of the form

\Delta_{h}^{n}f(x)=0

(30)

13.
- —
  
  Let us first examine some general properties of the solutions of equation (28). Let us still assume $0=\alpha_{0}<\alpha_{1}<\alpha_{2}<\ldots\ldots<\alpha_{n}$ and that $|f(x)|<\mathrm{M}$ In $\left(b^{\prime},b\right)$ . By taking $x+\alpha_{1}h=b^{\prime}$ , SO $h=\frac{b^{\prime}-x}{\alpha_{1}}$ And $x+\alpha_{n}h<b$ , we find

|f(x)|<\lambda\mathbf{M}\text{ pour }x>b^{\prime}-\rho\left(b-b^{\prime}\right)

Or $\lambda=\frac{\sum_{i=1}^{n}\left|a_{i}\right|}{\left|a_{0}\right|}$ And $\rho=\frac{\alpha_{1}}{a_{n}-\alpha_{1}}$ . We easily deduce that $|f(x)|<\lambda^{2}\mathrm{M}$ For $x>b^{\prime\prime}-\rho\left(b-b^{\prime\prime}\right)=b^{\prime}-\left[(1+\rho)^{2}-1\right]\left(b-b^{\prime}\right)|f(x)|<\lambda^{3}\mathrm{M}$ For $x>b^{\prime\prime\prime}-p\left(b-b^{\prime\prime\prime}\right)=b^{\prime}-\left[(1+p)^{3}-1\right]\left(b-b^{\prime}\right)$
………………………… $|f(x)|<\lambda^{s}M$ For $x>b^{\prime}-\left[(1+p)^{s}-1\right]\left(b-b^{\prime}\right)$ .
We successively posed $b^{\prime\prime}=b^{\prime}-p\left(b-b^{\prime}\right),b^{\prime\prime\prime}=b^{\prime\prime}-p\left(b-b^{\prime\prime}\right),\ldots$
If $s$ is the smallest positive integer such that $(1+p)^{s}>\frac{b-a}{b-b^{\prime}}$ , we have

|f(x)|<\lambda^{s}\mathbf{M}\text{ dans }(a,b)

and we can therefore state the following property:
Any solution of equation (28) bounded in a partial interval, as small as one wants, is bounded in the interval (a, b).

In the demonstration we assumed that the function is bounded in the interval ( $b^{\prime},b$ ), which does not restrict generality.

We see that:
Any solution of equation (28), which is identically zero in a subinterval, however small it may be, is identically zero in ( $\alpha,b$ ).

This property can also be stated in the following form:
If two solutions of equation (28) coincide in a subinterval, however small it may be, they coincide everywhere in ( $a,b$ ).

In particular:
Any solution of equation (28) which reduces to a polynomial in a subinterval, however small, is a polynomial of degree $k-1$ In ' $a,b$ ).

Let us further demonstrate the following lemma:
Any solution of equation (28) which is zero almost everywhere, is identically zero in the interval ( $a,b$ ).

Suppose there exists a point $x_{0}$ Or $f\left(x_{0}\right)\neq 0$ Let's vary $h$ between the limits $\rho\frac{b-x_{0}}{\alpha_{n}},\frac{b-x_{0}}{\alpha_{n}}$ Or $\rho=\max_{i=1,2,,,,n-1}\left(\frac{\alpha_{i}}{\alpha_{i+1}}\right)<1$ ; then the point $x_{i}=x_{0}+\alpha_{i}h$ describes the interval ( $x_{i}^{\prime},x_{i}^{\prime\prime}$ ) and these intervals are non-encroaching and do not contain the point $x_{0}\left({}^{2}\right)$ . At all points $x_{1}$ of ( $x_{1}^{\prime},x_{1}^{\prime\prime}$ ) corresponds, in the previous manner, to the points $x_{i},i>1$ in the intervals ( $x_{i}^{\prime},x_{i}^{\prime\prime}$ ) respectively. To all together $\mathrm{E}_{1}$ of points of ( $x_{1}^{\prime},x_{1}^{\prime\prime}$ ) corresponds to a set $\mathrm{E}_{i}$ of ( $r_{i}^{\prime},x_{i}^{\prime\prime}$ ) which is obtained from $\mathrm{E}_{1}$ by a similarity. Either $\mathrm{E}_{1}$ the whole, of measurement $x_{1}^{\prime\prime}-x_{1}^{\prime}$ , points of ( $x_{1}^{\prime},x_{1}^{\prime\prime}$ ) on which the function is zero. The relation $\sum_{i=0}^{n}a_{i}f\left(x_{i}\right)=0$ shows us that at all zero $x_{1}$ of $f(x)$ matches at least one $x_{i},i>1$ where the function is not zero. Or, in general, $\mathrm{E}_{1}^{(i)}$ the subset of $\mathrm{E}_{1}$ on which $f\left(x_{1}\right)=f\left(x_{2}\right)=\cdots=f\left(x_{i-1}\right)=0,f\left(x_{i}\right)\neq 0$ . We have

\mathrm{E}_{1}=\mathrm{E}_{1}^{(2)}+\mathrm{E}_{1}^{(3)}+\cdots+\mathrm{E}_{1}^{(n)}

(31)

But the whole $\mathrm{E}_{i}^{(i)}$ corresponding to $\mathrm{E}_{1}^{(i)}$ is of zero measure, it is therefore the same for $\mathrm{E}_{1}^{(i)}$ . Formula (31) is therefore absurd. The stated lemma is therefore completely demonstrated.

⁰⁰footnotetext: ⁽⁹⁾ These conditions are not, moreover, essential for the demonstration,

14. - The properties of the equation ( 30 are well known and have already been obtained almost all $\left({}^{3}\right)$ . We will recall these properties.

Any continuous solution of equation (30) is a polynomial of degree $n-1$ .

The quotient

\left[x_{0},x_{1},\ldots,x_{n};f\right]=\frac{U\left(x_{i},x_{1},\ldots,x_{n};f\right)}{V\left(x_{0},x_{1},\ldots,x_{n}\right)}

is the divided difference of order $n$ of the function $f(x)$ on the points $x_{0},x_{1},\ldots,x_{n}$ , always assumed to be distinct. Here $\mathrm{U}\left(x_{1},x_{1},\ldots,x_{n};f\right)$ is the determinant that we deduce from the Vandermonde determinant $\mathrm{V}\left(x_{0},x_{1},\ldots,x_{n}\right)$ by replacing the elements $x_{j}^{n},x_{i}^{n},\ldots,x_{n}^{n}$ by $f\left(x_{0}\right),f\left(x_{1}\right),\ldots,f\left(x_{n}\right)$ respectively. The difference in order $n$ , $\Delta_{h}^{n}f(x)$ is, up to a factor independent of the function, a divided difference,

\frac{\Delta_{h}^{n}f(x)}{n!h^{n}}=[x,x+h,\ldots,x+nh;f]

If the function $f(x)$ checks equation (30), we also have

\left[x+r_{0}h,x+r_{1}h,\ldots,x+r_{n}h;f\right]=0

(32)

$r_{0},r_{1},\ldots,r_{n}$ being rational numbers. This formula also results from a more general relation [10]. Consider a sequence of points $x_{0}<x_{1}<\cdots<x_{k}(k>n)$ . Any difference divided $\left[x_{i_{0}},x_{i_{1}},\ldots,x_{i_{n}};f\right]$ taken on $n+1$ of these points is of the form

	$\displaystyle{\left[x_{i_{0}},x_{i_{1}},\ldots,x_{i_{n}};f\right]=\sum_{i=0}^{k-n}\mathrm{\penalty 10000\ A}_{i}\left[x_{i},x_{i+1},\ldots,x_{i\vdash n};f\right]}$		(33)
	$\displaystyle A_{i}\geqq 0,i=0,1,\ldots,k-n,\sum_{i=0}^{k-n}A_{i}=1$

THE $A_{i}$ being independent of the function $f(x)$ . So, any difference divided $\left[x_{i_{0}},x_{i_{1}},\ldots,x_{i_{n}};f\right]$ is an arithmetic mean of the divided differences $\left[x_{i},x_{i+1},\ldots,x_{i+n};f\right],i=0,1,\ldots,k-n$ .
⁽⁵⁾ For equation (30) there are still other interesting problems which have not yet been solved but which we will not deal with in this work.

From formula (32) it follows that $f(x)$ reduces to a polynomial of degree $n-1$ on the set of points which rationally divide the interval ( $a,b$ ). The solution being assumed to be continuous, the stated property results. This property is a special case of a theorem of MLEJ Brouwer [2]. A direct demonstration of it was given by M. Th. Anghelutza [1].

Let us now demonstrate that:
Any bounded solution of equation (30) is continuous in the interval ( $a,b$ ).

So, indeed, $m>1$ a suitably chosen positive integer. We can write

[x,x+h,x+mh,x+2mh,\ldots,x+(n-1)mh;f]=0

The function being assumed to be bounded, we can find a $m$ large enough so that the absolute value of the second member is $<\varepsilon$ , and this whatever $\varepsilon>0$ . We can then find a positive number $\eta$ such that we have $a<x+(n-1)mh<b$ For $|h|<\eta$ . Formula (34) is then effectively applicable and gives us

|f(x)-f(x+h)|<\varepsilon\text{ pour }\quad|h|<\eta

which proves the continuity ( ⁴ ).
We have the following theorem:
Any bounded solution of equation (30) is a polynomial of degree $n-1$ .

This theorem is a special case of a more general theorem of MA Marchaud [8], which generalizes that of MLEJ Brouwer.

⁰⁰footnotetext: ( ¹ ) We can also choose the numbers appropriately

m

And

\eta

so that we have

\AA

being a constant independent of

l4

We therefore also have the following theorem:
The general bounded solution of a reducible equation of order $k$ is any polynomial of degree $k-1$ .

The various properties that we have established have been studied, for the equation $\Delta_{h}^{2}f(x)=0$ , by Darboux [4, 9]. Darboux deals with the Cauchy equation $f(x+y)=f(x)+f(y)$ in the meantime $(-\infty,+\infty)$ . To demonstrate the equivalence of the two problems, it is first necessary to prove, which is not entirely obvious, that any solution to the equation $\Delta_{h}^{2}f(x)=0$ In $(a,b)$ is made up of the values in ( $a,b$ ) of a solution of this same equation considered in the interval ( $-\infty,+\infty$ ). We can assume the interval ( $a,b$ ), closed and then either $x$ a point outside ( $a,b$ ) And $x_{0}$ a point of ( $a,b$ ) which rationally divides the interval $(a,x)$ . We define the value of the function at the point $x$ by equality $\left[a,x_{0},x;f\right]=0$ . In this way $f(x)$ is completely determined. We easily verify that the function thus defined verifies the equation

\Delta_{h}^{2}f(x)=0,\text{ dans }(-\infty,+\infty)

(35)

This property also tells us that any solution to equation (35) is completely determined as soon as we know its values in an interval, however small it may be. We see immediately that if $f(x)$ is a solution of equation (35), the function $f(x)-f(0)$ verifies the Cauchy equation. Darboux also demonstrated that it is sufficient for the function to be bounded above (or below) to draw the conclusion that it reduces to a polynomial. This property is no longer true for $n>2$ .
15. - The study of the general equation (28) comes back to the study of equations of the form (29). From this last equation we deduce

\sum_{i_{1}=0}^{r_{1}}\sum_{i_{2}=0}^{r_{2}}\ldots\sum_{i_{n}\equiv 0}^{r_{n}}\delta_{h}^{\left(\alpha_{i}\right)}f\left[x+\left(i_{1}\alpha_{1}+i_{2}\alpha_{2}+\cdots+i_{h}\alpha_{h}\right)h\right]=0,

which can also be written

\delta_{h}^{\left(r_{i}a_{i}\right)}f(x)=0,

(36)

Or $r_{1},r_{2},\ldots,r_{n}$ are positive integers. Any solution of equation
(29) also verifies any equation (36) where $r_{1},r_{2},\ldots,r_{n}$ are rational numbers, positive or negative.

Now consider a continuous solution of equation (29). We can always choose rational numbers $r_{1},r_{2},\ldots,r_{n}$ such as $r_{1}\alpha_{1},r_{2}\alpha_{2},\ldots,r_{n}x_{n}$ are as close as we want to 1. From the continuity then results that:

Any continuous solution of equation (29) also satisfies equation (29) in which we assume $\alpha_{i}=1,i=1,2,\ldots,n$ , so equation (30).

We can therefore state the following theorem:
Any continuous solution of equation (28) is a polynomial of degree k-1.

We can easily demonstrate that this result still holds if we assume the function to be summable in the interval $(a,b)$ .

Either $f(x)$ a summable solution of equation (28). We know that the indefinite integral $g(x)=\int_{a}^{x}f(x)dx$ is a continuous function and derivable almost everywhere. Moreover we have $g^{\prime}(x)=f(x)$ almost everywhere. We can easily see that if $f(x)$ verifies equation (28), the function $g(x)$ check the equation

\sum_{i=0}^{n}a_{i}\left[g\left(x+\left(\alpha_{i}+1\right)h\right)-g\left(x+\alpha_{i}h\right)\right]=0

of order $k+1$ . It follows that $g(x)$ is a polynomial of degree $k$ , so that $f(x)$ coincides almost everywhere with a polynomial $\mathbf{P}(x)$ of degree $k-1$ . The difference $f(x)-\mathrm{P}(x)$ verifies equation (28) and is zero almost everywhere, so we can state, by virtue of the lemma of No. 13, the following theorem:

Every summable solution of equation (28) is a polynomial of degree k-1.
16. - Let us now prove the following lemma:

Any measurable solution of equation (28) is bounded in the interval ( $a,b$ ).

Let us assume the opposite. So let $f(x)$ an unbounded measurable solution. According to the results of No. 13, this function is not bounded in any subinterval. In any subinterval there is therefore at least one point $\xi$ Or $|f(\xi)|>A$ , whatever the positive number A.

Let us always assume $0=\alpha_{0}<\alpha_{1}<\ldots<\alpha_{n}$ . Either $c$ the middle of the interval $(a,b)$ And $a_{1},b_{1}$ the midpoints of the intervals $(a,c)(c,b)$ .

Given a positive number A , it exists in the interval ( $a_{1},b_{1}$ ) a point $\xi$ Or

|f(\bar{\xi})|>\frac{\left|a_{1}\right|+\left|a_{2}\right|+\cdots+\left|a_{n}\right|}{\left|a_{0}\right|}\mathrm{A}

(37)

Let's take $\max_{i=1,2,\ldots,n-1}\left(\frac{\alpha_{i}}{\alpha_{i+1}}\right)=\rho<1$ and be $h^{\prime}=\frac{(b-a)}{4x_{n}},h^{\prime\prime}=-\frac{b-a}{4x_{n}}x_{i}^{\prime}=\xi+\frac{\alpha_{i}\rho}{4\alpha_{n}}(b-a),x_{i}^{\prime\prime}=\xi+\frac{\alpha_{i}(b-a)}{4\alpha_{n}},i=1,2,\ldots,n$ . The intervals $\left(x_{i}^{\prime},x_{i}^{\prime\prime}\right)$ are then non-encroaching and do not contain the point $\xi$ . Either $h$ a number between $h^{\prime}$ And $h^{\prime\prime}$ . By posing $x_{i}=\xi+\alpha_{i}h$ , the point $x_{i}$ is in the (open) interval $\left(x_{i}^{\prime},x_{i}^{\prime\prime}\right)$ and equality

f(\xi)=-\frac{\sum_{i=1}^{n}a_{i}f\left(x_{i}\right)}{a_{0}}

shows us that we must have $\left|f\left(x_{i}\right)\right|>\mathrm{A}$ for at least one value of $i$ . Otherwise inequality (37) would be impossible. The points $x_{i}$ correspond by similarity in the intervals ( $x_{i}^{\prime},x_{i}^{\prime\prime}$ ). Either $\mathrm{E}_{1}$ the set of points in the interval ( $x_{1}^{\prime},x_{1}^{\prime\prime}$ ) where we have $\left|f\left(x_{1}\right)\right|>\mathrm{A}$ . Either $\mathrm{E}_{2}$ the set of points of ( $x_{1}^{\prime},x_{1}^{\prime\prime}$ ) Or $\left|f\left(x_{1}\right)\right|\leqq\mathrm{A},\left|f\left(x_{2}\right)\right|>\mathrm{A}$ , the points $x_{1},x_{2}$ being two corresponding points in $\left(x_{1}^{\prime},x_{1}^{\prime\prime}\right),\left(x_{2}^{\prime},x_{2}^{\prime\prime}\right)$ and either $\mathrm{E}_{2}^{\prime}$ all of these points $x_{2}$ . Generally speaking, either $\mathrm{E}_{i}$ all the points of $\left(x_{1}^{\prime},x_{1}^{\prime\prime}\right)$ Or $\left|f\left(x_{1}\right)\right|\leqq\mathrm{A},\left|f\left(x_{2}\right)\right|\leqq\mathrm{A},\ldots,\left|f\left(x_{i-1}\right)\right|\leqq\mathrm{A},\left|f\left(x_{i}\right)\right|>\mathrm{A}$ , the points $x_{1},x_{2},\ldots,x_{i}$ being corresponding points in the intervals $\left(x_{1}^{\prime},x_{1}^{\prime\prime}\right),\left(x_{2}^{\prime},x_{2}^{\prime\prime}\right),\ldots,\left(x_{i}^{\prime},x_{i}^{\prime\prime}\right)$ and either $\mathrm{E}_{i}^{*}$ all of these points $x_{i}$ . All these sets are measurable, by virtue of the well-known properties of measure and measurable functions. By denoting by $|\mathrm{E}|$ the measure of the set E, we have

\left|E_{1}\right|+\left|E_{2}\right|+\cdots+\left|E_{n}\right|=x_{1}^{\prime\prime}-x_{1}^{\prime}=\frac{\alpha_{1}(b-a)(1-\rho)}{4x_{n}}

since

\mathrm{E}_{1}+\mathrm{E}_{2}+\cdots+\mathrm{E}_{n}=\text{ l'intervalle ouvert }\left(x_{1}^{\prime},x_{1}^{\prime\prime}\right)

But, the measure of the set on which we have $|f(x)|>\mathrm{A}$ is at least $\left|E_{1}\right|+\left|E_{2}^{*}\right|+\cdots+\left|E_{n}^{*}\right|$ . Gold, $\left|E_{i}^{*}\right|=\frac{\alpha_{i}}{\alpha_{1}}\left|E_{i}\right|>\left|E_{i}\right|$ , so we have

||f(x)|>\mathrm{A}|\geqq\frac{\alpha_{1}(b-a)(1-\rho)}{4x_{n}}=\text{ nombre positif fixe, }

the first member designating the measure of the set of $x$ for which $|f(x)|>\mathrm{A}$ . This inequality being true whatever A , the function cannot be measurable, by virtue of a theorem of ME Borel (5). This contradiction demonstrates the stated lemma.

Any measurable and bounded function is summable, so we finally have the following theorem:

The general measurable solution of equation (28), of order $k$ , is any polynomial of degree k-1.

This is the generalization of the theorem of MW Sierpinske, who considered equation (30) for $n=2$ [12]. We have already given this property for equation (30) and $n$ any [10]. The previous demonstration is also analogous to that of MW Sierpinski (6).
17. - We have assumed up to now that the function $f(x)$ be defined in an interval ( $a,b$ ). For the rest, it is interesting to also consider a slightly more general case. Let us suppose the function defined on a set E , contained in $(a,b)$ and measurement b.-a. Equation (28) must then be satisfied for all values of $x$ And $h$ such as $x+\alpha_{i}h,i=0,1,\ldots,n$ belong to the set E. Let us further suppose $0=\alpha_{0}<\alpha_{1}<\cdots<\alpha_{n}$ and we make the remark on the structure of equation (28) that if the point $x$ belongs to the set E , all points $x+\alpha_{i}h,i=1$ , $2,\ldots,n$ also belong to E for almost all values of $h$ checking the equalities $a<x+\alpha_{i}h<b,i=1,2,\ldots,n$ . The demonstration is immediate. The number $h$ varies in the interval $\left(-\frac{x-a}{\alpha_{n}},\frac{b-x}{\alpha_{n}}\right)$ . Either $e_{i}$ all of the $h$ for which $x+\alpha_{i}h$ does not belong to E. The set $e_{i}$ are of zero measure, so it is the same for their sum and the property results from it. We also see that for all $h$ (such as $\left|\alpha_{n}h\right|<b-a$ ), the points $x+\alpha_{i}h,i=0,1,\ldots,n$ belong to E for almost all values of $x$ checking the inequalities $a<x+\alpha_{i}h<b,i=0,1,\ldots,n$ .

We now have the following properties:
Any function, defined on E , which satisfies equation (28) and which is zero almost everywhere, is identically zero on E .

The demonstration is analogous to that given in the case where E is an interval (No. 13).

⁰⁰footnotetext:

\left({}^{5}\right.

) According to this theorem if

f(x)

is measurable, at all

e>0

corresponds to a number A such that we have

|f(x)>\mathrm{A}|<\epsilon,|f(x)<-\mathrm{A}|<\epsilon

.
⁽⁸⁾ Regarding the Cauchy equation, as well as the bibliography on this subject, see the work of MW Serrpingki in volume I of Fundamenta Mathematicae.

Any solution of equation (28), summable over E , is a polynomial of degree $k-1$ .

Let us again consider the indefinite integral $g(x)=\int_{a}^{x}f(x)dx$ , which is a continuous function of $x$ in the meantime $(a,b)$ . We can easily see that the function $g(x)$ check the equation again

\sum_{i=0}^{n}a_{i}\left[g\left(x+\left(\alpha_{i}+1\right)h\right)-g\left(x+\alpha_{i}h\right)\right]=0

in the meantime ( $a,b$ ) The rest of the demonstration is done exactly as in No. 15. All this succeeds because of the property of being able to neglect sets of zero measure in integration in the sense of Mr. Lebesgue.

Any solution to equation (28), which is bounded on the part of E belonging to a subinterval of ( $a,b$ ), however small it may be, is bounded on E.

\left.|f(x)|<\lambda\mathrm{M}\text{ pour }\mathrm{E}=x>b^{\prime}-\rho b-b^{\prime}\right),\ldots\text{ etc. }

Any measurable solution of equation (28) is bounded on $E$ . The demonstration of this property is done as in No. 16.
Finally, the final theorem of the previous No. still remains:
The general measurable solution of equation (28), on the set E, is any polynomial of degree k-1.

We demonstrate this exactly as above.
In summary, all the properties studied for the case of an interval remain true if we exclude from this interval a set of zero measure.

This extension was successful because of the well-known properties of measure-zero sets and especially because of the property that any subset of a measure-zero set is still measurable and of measure-zero.

CHAPTER III

On pseudo-polynomials of two or more variables

18.
- —
  
  We consider functions $f\left(x_{1},x_{2},\ldots,x_{m}\right.$ real, of $m$ real variables $x_{1},x_{2},\ldots,x_{m}$ , uniform and defined in a bounded and open domain D. The point ( $x_{1},x_{2},\ldots,x_{m}$ ) is related to a coordinate axis system $\mathrm{O}x_{1}x_{2}\ldots x_{m}$ , which we can assume to be rectangular. Let $D_{1}$ a hyperparallelipede completely interior to D (therefore all the points of the closed domain $D_{1}$ belong to $D$ ) and having its faces parallel to the coordinate hyperplanes. To any point P of D corresponds a sequence of hyperparallelipipeds $\mathrm{D}_{1},\mathrm{D}_{2},\ldots,\mathrm{D}_{s}$ enjoying the following properties:
19.

all the $D_{i}$ have their respective faces parallel and are completely interior to D.
$2^{0}$ . open areas $D_{i},D_{i+1}$ have a common part, $i=1,2,\ldots,s-1$ .
20.

the point $\mathbf{P}$ is inside $\mathrm{D}_{s}$ .

We denote by R the minimum hyperparallelipiped containing $D$ and having its faces parallel to the hyperplanes of the coordinates. R is therefore the domain defined by the inequalities $a_{i}<x_{i}<b_{i}$ , $i=1,2,\ldots,m$ . Most often we can also assume that D coincides with $R$ and we can then also assume, without great inconvenience, that this domain is closed.

The previous properties remain valid if we take instead of the axes $\mathrm{O}x_{1}x_{2}\ldots x_{m}$ , a new axis system $\mathrm{O}x^{\prime}{}_{1}x^{\prime}{}_{2}\ldots x^{\prime}{}_{m}$ , forming a real $m$ -hedron. In Chapter V we will make extensive use of changes of axes.

To better understand the questions that follow, we will start with the properties of pseudo-polynomials of two variables.
19. - Let us recall the definition of the partial divided differences of the function $f(x,y)$ . Let us call, with MA Marchaud [8], a network of order ( $m,n$ ) a system formed by $m+1$ lines parallel to the axis $\mathrm{O}y$ And $n+1$ lines parallel to the axis $\mathrm{O}x$ . We will assume that the lines forming the network are distinct. Any line in the network is characterized by its abscissa if it is parallel to the Oy axis and by its ordinate if it is parallel to the axis $\mathrm{O}x$ , More
explicitly we can designate a network of order ( $m,n$ ) by

\left(x_{0},x_{1},\ldots,x_{m}\mid y_{0},y_{1},\ldots,y_{n}\right),

(38)

highlighting the abscissas and ordinates of the component lines. The lines that form the network (38) intersect at $\mathrm{N}=(m+1)(n+1)$ points which are the nodes of this network. It goes without saying that in the following it is sufficient to consider only the parts of the component lines of the network, included in the domain $D$ considered in each problem. In particular, we only consider networks whose nodes are included in $D$ .

Consider the network (38) and denote by $\mathbf{M}_{1},\mathbf{M}_{2},\ldots,\mathbf{M}_{\mathrm{N}}$ the nodes of this network, therefore the points $\left(x_{i},y_{j}\right),i=0,1,\ldots,m$ , $j=0,1,\ldots,n$ . Let us designate by $\mathrm{V}_{m,n}\left(\mathrm{M}_{1},\mathrm{M}_{2},\ldots,\mathrm{M}_{\mathrm{N}}\right)$ the order determinant $\mathbf{N}$ whose general line is formed by the elements $x_{i}^{r}y_{j}^{s},r=0,1,\ldots,m,s=0,1,\ldots,n$ and either $\mathrm{U}_{m,n}\left(\mathrm{M}_{1},\mathrm{M}_{2},\ldots,\mathrm{M}_{\mathrm{N}};f\right)$ the determinant that we deduce from the previous one by replacing the elements $x_{i}^{m}y_{j}^{n}$ by $f\left(x_{i},y_{j}\right)$ respectively, $i=0,1,\ldots,m,j=0,1,\ldots,n$ . By definition, the partial divided difference of order ( $m,n$ ) of the function $f(x,y)$ on the points $\mathrm{M}_{1},\mathrm{M}_{2},\ldots,\mathrm{M}_{\mathrm{N}}$ , or on the network (38), is equal to the quotient

\underline{\mathrm{U}}_{m,n}\left(\mathrm{M}_{1},\mathrm{M}_{2},\ldots,\mathrm{M}_{\mathrm{N}};f\right)

(39)

which does make sense since the denominator is $\neq 0$ . Changing our previous notation [10] a little, we will designate expression (39) by

\left[x_{0},x_{1},\ldots,x_{m}\mid y_{0},y_{1},\ldots,y_{n};f\right]

We can put this divided difference in a form that allows us to better see its structure. Let $\phi(x)=\left(x-x_{0}\right)\left(x-x_{1}\right)\ldots\left(x-x_{m}\right)$ , $\psi(y)=\left(y-y_{0}\right)\left(y-y_{1}\right)\ldots\left(y-y_{n}\right)$ , We have

\left[x_{0},x_{1},\ldots,x_{m}\mid y_{0},y_{1},\ldots,y_{n};f\right]=\sum_{i=0}^{m}\sum_{j=0}^{n}\frac{f\left(x_{i},y_{j}\right)}{\phi^{\prime}\left(x_{i}\right)\psi^{\prime}\left(y_{j}\right)}.

(40)

This form fully justifies the name partial divided difference because it is only a superposition of divided differences taken successively with respect to the variables $x,y$ . If we pose $x+\alpha_{i}h_{1}$ And $y+\beta_{j}h_{2}$ instead of $x_{i}$ And $y_{i}$ , the expression is, up to a factor independent of the function, of the form (14).

The difference in order ( $m,n$ ), $\Delta_{h_{1},h_{2}}^{m,n}f(x,y)$ is, up to a factor independent of the function, a divided difference of order ( $m,n$ ) $\frac{1}{m!n!h_{1}^{m}h_{2}^{n}}\Delta_{h_{1},h_{2}}^{m,n}f(x,y)=\left[x,x+h_{1},\ldots,x+mh_{1}\mid y,y+h_{2},\ldots,y+nh_{2};f\right]$ .

The corresponding network is formed by equidistant lines.
If we do $h_{1}=h_{2}=h$ , the previous difference becomes $\Delta_{h}^{m,n}f(x,y)$ so a difference of order ( $m,n$ ) of the first kind. We can say that every difference of the second kind contains a difference of the first kind. In this way, some of the properties that we will state later for differences of the first kind remain a fortiori true for those of the second kind.

Consider the network

\left(x_{0},x_{1},\ldots,x_{m_{1}}\mid y_{0},y_{1},\ldots,y_{n_{1}}\right)

(41)

of order ( $m_{1},n_{1}$ ), Or $m_{1}\geqq m,n_{1}\geqq n,m_{1}+n_{1}>m+n$ . The form (40) of the partial divided difference shows us that:

Any partial divided difference of order ( $m,n$ ), taken on $(m+1)(n+1)$ points chosen among the nodes of the network (41), is an arithmetic mean of the partial divided differences

\begin{gathered}{\left[x_{i},x_{i+1},\ldots,x_{i+m}\mid y_{j},y_{j+1},\ldots,y_{j+n};f\right],}\\ i=0,1,\ldots,m_{1}-m,j=0,1,\ldots.n_{1}-n.\end{gathered}

Partial divided differences were defined with respect to the Oxy coordinate axes. Taking a new axis system $Ox^{\prime}y^{\prime}$ we define in the same way the networks and the partial divided differences with respect to these axes. We can immediately write these divided differences, for example under the foritie (40), which is particularly convenient.
20. - Let us call, still according to MA Marghaud [8], pseudopolynomial of order ( $m,n$ ), any expression of the form

\sum_{i=0}^{m}x^{i}\mathrm{\penalty 10000\ A}_{i}(y)+\sum_{j=0}^{n}y^{j}\mathrm{\penalty 10000\ B}_{j}(x),

(42)

Or $\mathrm{A}_{i}(y)$ are functions of $y$ alone and $\mathrm{B}_{j}(x)$ functions of $x$ alone (of the form indicated at the beginning of No. 12). We will call them the coefficients of the pseudo-polynomial. We agree to call pseudo-polynomial of order ( $-1,n$ ) a polynomial of degree $n$ in $\mathscr{Y}$ ,
pseudo-polynomial of order ( $m,-1$ ) a polynomial of degree $m$ in $x$ and pseudo-polynomial of order ( $-1,-1$ ) the identically zero function.

The partial divided difference of order ( $m,n$ ) of a pseudopolynomial of order ( $m-1,n-1$ ) is identically zero.

We immediately deduce that:
A pseudo-polynomial of order ( $m,n$ ) is completely determined if we know its values on a network of order (m, n).

In the study of pseudo-polynomials it is sufficient to assume that the domain D reduces to the rectangle R . This also results from a kind of extension property which is more or less obvious. Let us assume that $f(x,y)$ be a pseudo-polynomial of order ( $m,n$ ) in each of the rectangles $D_{1},D_{2}$ . If these (open) domains have a common part, it immediately follows that $f(x,y)$ is a pseudo-polynomial of order ( $m,n$ ) in the domain formed by the union of the rectangles $D_{1},D_{2}$ . To see this, simply take a network of order ( $m,n$ ) whose nodes belong to the common part of $D_{1},D_{2}$ .

We also have the converse property:
Any function $f(x,y)$ whose partial divided difference of order ( $m,n$ ) is zero identically is a pseudo-polynomial of order ( $m-1,n-1$ ).

Some of the properties of the pseudo-polynomial are reflected in its coefficients. Thus

If the pseudo-polynomial (42) is bounded in the rectangle R, its coefficients are functions bounded in the intervals $\left(a_{1},b_{1}\right)\left(a_{2},b_{2}\right)$ respectively.

Suppose that for the pseudo-polynomial $f(x,y)$ , of order ( $m,n$ ), we have $|f(x,y)|<M$ . Let's give to $x,m+1$ distinct values $x_{0},x_{1},\ldots,x_{m}$ and write the system

\sum_{i=0}^{m}x_{r}^{i}A_{i}(y)+\sum_{j=0}^{n}y^{j}B_{j}\left(x_{r}\right)=f\left(x_{r},y\right),\quad r=0,1,\ldots,m.

(43)

We can determine a number $\mathbf{M}^{\prime}$ such that we have

\left|f\left(x_{r},y\right)-\sum_{j=0}^{n}y^{j}\mathrm{\penalty 10000\ B}_{j}\left(x_{r}\right)\right|<\mathrm{M}^{\prime},\quad r=0,1,\ldots,m,

regardless of $y$ . Solving the system (43) with respect to the coefficients $\mathrm{A}_{i}(y)$ , we deduce the property stated for these coefficients.

We proceed in the same way to show that the $B_{j}(x)$ are bounded. This property is not true, as are the following ones, for pseudo-polynomials of more than two variables. We will see a little later how they must be modified. From the relation

\left[x,x_{0},x_{1},\ldots,x_{m}\mid y,y_{0},y_{1},\ldots,y_{n};f\right]=0

we also deduce that we can write a pseudo-polynomial of order ( $m,n$ ) in the following form

	$\displaystyle f(x,y)$	$\displaystyle=\sum_{i=0}^{m}\frac{\phi x)}{\left(x-x_{i}\right)\phi^{\prime}\left(x_{i}\right)}+\sum_{j=0}^{n}\frac{\psi(y)}{\left(y-y_{j}\right)\psi^{\prime}\left(y_{j}\right)}+$		(44)
		$\displaystyle+\sum_{i=0}^{m}\sum_{j=0}^{n}\frac{\phi(x)\psi(y)f\left(x_{i},y_{j}\right)}{\left(x-x_{i}\right)\left(y-y_{j}\right)\phi^{\prime}\left(x_{i}\right)\psi^{\prime}\left(y_{j}\right.}$

Or $\left.\phi(x)=\left(x-x_{0}\right)\left(x-x_{1}\right)\ldots\left(x-x_{m}\right),\quad\psi\cdot y\right)=\left(y-y_{0}\right)\left(y-y_{1}\right)\ldots\left(y-y_{n}\right)$ . The corresponding coefficients of two identical pseudo-polynomials differ only by polynomials in $x$ And $y$ . The previous property thus results from the simple inspection of formula (44). We also see that:

If a pseudo-polynomial is continuous its coefficients are continuous functions.

The converse of this property is obviously true.
Let us say that a function is linearly measurable if it is measurable with respect to each of the variables $x$ And $y$ separately. A measurable function $f(x,y)$ is not, in general, linearly measurable but, according to a theorem of MG Fubini [6], any measurable function is a measurable function of $x$ for almost all $y$ and a measurable function of $y$ for almost all $x$ . Now consider a measurable pseudo-polynomial. This pseudo-polynomial is measurable on $m+1$ parallel to the axis $Oy$ and on $n+1$ parallel to the axis $\mathrm{O}x$ and we deduce that:

The coefficients of a measurable pseudopolynomial are measurable functions.

Conversely:
If the coefficients of a pseudo-polynomial are measurable, this pseudo-polynomial is measurable.

Indeed, if $\phi(x)$ And $\psi(y)$ are measurable functions, their product $\phi(x)\psi(y)$ is a (superficially) measurable function with respect to $x$ And $y$ .

Mathematica, vol. XIV.

Moreover, any measurable pseudo-polynomial is linearly measurable and vice versa. It is also clear that if a pseudopolynomial is B measurable, its coefficients are B measurable and vice versa.
21. - The lines of the network (41) divide rationally if the points $x_{0},x_{1},\ldots,x_{m}$ as well as the points $y_{0},y_{1},\ldots,y_{n}$ divide rationally. It is now clear what is meant by a network that divides rationally and is everywhere dense in the rectangle R. The lines of the network parallel to the axis $\mathrm{O}y$ respectively to the axis $\mathrm{O}x$ divide rationally and have dense abscissas and ordinates respectively in the intervals $\left(a_{1},b_{1}\right),\left(a_{2},b_{2}\right)$ . Of course, such a network is not of a determined order. It is an infinite network and more precisely a doubly infinite network. We then have the following property:

If a difference of order ( $m,n$ ) of the first kind of function $f(x,y)$ is zero identically in R , we can construct a network which divides rationally and which is everywhere dense in R such that any partial divided difference of order ( $m,n$ ), taken on $(m+1)(n+1)$ points chosen from the nodes of this network, or zero.

The everywhere dense network can be constructed in the following way. The abscissas of the lines parallel to $\mathrm{O}y$ are $a_{1}+h$ , Or $h$ is rational ( $>0$ ) and the ordinates of the lines parallel to $\mathrm{O}x$ are $a_{2}+h^{\prime}$ , Or $h^{\prime}$ is rational ( $>0$ ). The numbers $h$ And $h^{\prime}$ are chosen such that we have $a_{1}+h<b_{1},a_{2}+h^{\prime}<b_{2}$ . The property results from the fact that the function reduces to a pseudo-polynomial of order ( $m-1,n-1$ ) on any set formed by the nodes of a network which divides rationally.

The general form of a difference of the first kind is in reality

\sum_{i=0}^{m}\sum_{j=0}^{n}(-1)^{m+n-i-j}\binom{m}{i}\binom{n}{j}f(x+ih,y+j\alpha h)

We assumed $\alpha=1$ , which does not restrict the generality. Indeed, we return to this case by a simple transformation (dilation) made on the variable $y$ .

We immediately deduce that:
If a difference of order ( $m,n$ ) of the first kind of the continuous function $f(x,y)$ is identically zero, the
partial divided difference of order ( $m,n$ ) is identically zero in D. The function is therefore reduced in D to a pseudo-polynomial of order $(m-1,n-1)$ .

This property results from the fact that given a network of order ( $m,n$ ), we can construct another network that divides rationally such that the corresponding nodes of these two networks are as close as we want.

Consider the functional equation of the second kind

\Delta_{h_{1},h_{2}}^{m,n}f(x,y)=0

(45)

in the field $D$ , which we can assume coincides with $R$ We have the following theorem, due to MA Marchaud [8]:

Any bounded solution of equation (45) is a pseudo-polynomial of order ( $m-1,n-1$ ) In $\mathbf{R}$ .

The demonstration is immediate. It suffices to demonstrate that if $f(x,y)$ is zero on a network of order ( $m-1,n-1$ ), it is identically null. Posing

g(x)=\sum_{j=0}^{n}(-1)^{n-j}\binom{n}{j}f\left(x,y+jh_{2}\right)

we have $\Delta_{h_{1}}^{m}g(x)=0$ , SO $g(x)$ is a polynomial of degree $m-1$ in $x$ who, being zero for $m$ values of $x$ , is identically zero. The relation $g(x)\equiv 0$ shows us that $f(x,y)$ is a polynomial of degree $n-1$ in $y$ who, being zero for $n$ values of $y$ , is identically zero.

The preceding statement is, of course, also valid for any domain D.
22. - The questions set out above can be extended to functions of any number of variables.

A network of order $\left(n_{1},n_{2},\ldots,n_{m}\right)$ is a system of $n_{i}+1$ hyperplanes parallel to the coordinate hyperplane $\mathrm{O}x_{1}\ldots x_{i-1}x_{i+1}\ldots x_{m}$ , $i=1,2,\ldots,m$ . Any hyperplane of the network is characterized by the abscissa of its point of intersection with the $m^{\text{dme }}$ coordinate axis. Highlighting the abscissas of the constituent hyperplanes, we can designate a network of order ( $n_{1},n_{2},\ldots,n_{m}$ ) by

\left(x_{10},x_{11},\ldots,x_{1n_{1}}\left|x_{20},x_{21},\ldots,x_{2n_{2}}\right|\ldots\mid x_{m0},x_{m1},\ldots,x_{mn_{m}}\right)

(46)

THE $\left(n_{1}+1\right)\left(n_{2}+1\right)\ldots\left(n_{m}+1\right)$ points $\left(x_{1i_{1}},x_{2i_{2}},\ldots,x_{mi_{m}}\right),i_{1}=0,1,\ldots,i_{1}$ , $i_{2}=0,1,\ldots,n_{2},\ldots,i_{m}=0,1,\ldots,n_{m}$ are the nodes of the network (46).

The partial divided difference of order $\left(n_{1},n_{2},\ldots,n_{m}\right)$ on the network (46) has a definition similar to that for two variables. We will designate this difference divided by
(47) $\left[x_{10},x_{11},\ldots,x_{1n_{1}}\left|x_{20},x_{21},\ldots,x_{2n_{2}}\right|\ldots\mid x_{m0},x_{m1},\ldots,x_{mn_{m}};f\right]$
and if we ask $\phi_{j}(x)=\left(x-x_{j0}\right)\left(x-x_{j1}\right)\ldots\left(x-x_{jn_{j}}\right)$ , the divided difference (47) can also be written

\sum_{i_{1}=0}^{n_{1}}\sum_{i_{2}=0}^{n_{2}}\ldots\sum_{i_{m}=0}^{n_{m}}\frac{f\left(x_{1i_{1}},x_{2i_{2}},\ldots,x_{mi_{m}}\right)}{\phi_{1}^{\prime}\left(x_{1i_{1}}\right)\phi_{2}^{\prime}\left(x_{2i_{2}}\right)\ldots\phi_{m}^{\prime}\left(x_{mi_{m}}\right)}

(48)

which fully justifies the name of partial divided difference.

The difference in order ( $n_{1},n_{2},\ldots,n_{m}$ ) is, up to a factor independent of the function, a partial divided difference of order $\left(n_{1},n_{2},\ldots,n_{m}\right)$ ,

\begin{gathered}\frac{1}{n_{1}!n_{2}!\ldots n_{m}!h_{1}^{n_{1}}h_{2}^{n_{2}}\ldots h_{m}^{n_{m}}}\Delta_{h_{1}}^{n_{1}},n_{2},\ldots,h_{2},\ldots,h_{m}f\left(x_{1},x_{2},\ldots,x_{m}\right)=\\ =\left[x_{1},x_{1}+h_{1},\ldots,x_{1}+n_{1}h_{1}\left|x_{2},x_{2}+h_{2},\ldots,x_{2}+n_{2}h_{2}\right|\ldots\right.\\ \left.\ldots\mid x_{m},x_{m}+h_{m},\ldots,x_{m}+n_{m}h_{m};f\right]\end{gathered}

The corresponding network is therefore formed by equidistant hyperplanes.

We still have the property:
The entire partial divided difference of order ( $n_{1},n_{2},\ldots,n_{m}$ ), taken on $\left(n_{1}+1\right)\left(n_{2}+1\right)\ldots\left(n_{m}+1\right)$ points chosen from the nodes of the network

\begin{gathered}\left(x_{10},x_{11},\ldots,x_{1k_{1}}\left|x_{20},x_{21},\ldots,x_{2k_{2}}\right|\ldots\mid x_{m0},x_{m1},\ldots,x_{mk_{m}}\right)\\ k_{1}\geq n_{1},k_{2}\geq n_{2},\ldots,k_{m}\geq n_{m},k_{1}+k_{2}+\cdots+k_{m}>n_{1}+n_{2}+\cdots+n_{m}\end{gathered}

is an arithmetic mean of the divided differences
$\left[x_{1i_{1}},x_{1i_{1}+1},\ldots,x_{1i_{1}+n_{1}}\left|x_{2i_{2}},x_{2i_{2}+1},\ldots,x_{2i_{2}+n_{2}}\right|\ldots\mid x_{mi_{m}},x_{mi_{m}+1},\ldots,x_{mi_{m}+n_{m}};f\right]$

i_{1}=0,1,\ldots,k_{1}-n_{1},i_{2}=0,1,\ldots,k_{2}-n_{2},\ldots,i_{m}=0,1,\ldots,k_{m}-n_{m}

We can define networks and partial divided differences with respect to any system of axes $\mathrm{O}x_{1}^{\prime}x_{2}^{\prime}\ldots x_{m}^{\prime}$ . We can immediately write these differences 1 divided for example in the form (48).
23. - A pseudo-polynomial of order ( $n_{1},n_{2},\ldots,n_{m}$ ) is an expression of the form

\sum_{j=1}^{m}\sum_{i=0}^{n_{j}}x_{j}^{i}\mathrm{\penalty 10000\ A}_{ji}\left(x_{1},x_{2},\ldots,x_{j-1},x_{j+1},\ldots,x_{m}\right)

where the coefficients $\mathbf{A}_{ji}$ are functions depending on variables $x_{1},x_{2},\ldots,x_{j-1},x_{j+1},\ldots,x_{m}$ alone.

If in order ( $n_{1},n_{2},\ldots,n_{m}$ ) we have $n_{j}=-1$ we agree that the $\mathrm{A}_{ii},i=0,1,\ldots,n_{j}$ , are all identically zero. In particular, the pseudo-polynomial of order '( $-1,-1,\ldots,-1$ ) is the identically zero function.

We have the following properties:
The partial divided difference of order ( $n_{1},n_{2},\ldots,n_{m}$ ) of a pseudo-polynomial of order ( $n_{1}-1,n_{2}-1,\ldots,n_{m}-1$ ) is identically zero.

A pseudo-polynomial of order ( $n_{1},n_{2},\ldots,n_{m}$ ) is completely determined if we know its values on a network of order $\left(n_{1},n_{2},\ldots,n_{m}\right)$ .

In the study of pseudo-polynomials we can assume that $D$ coincides with a hyperparallelipede R. The explanation is as in the case of two variables.

Any function $f\left(x_{1},x_{2},\ldots,x_{m}\right)$ whose partial divided difference of order ( $n_{1},n_{2},\ldots,n_{m}$ ) is zero identically, is a pseudo-polynomial of order ( $n_{1}-1,n_{2}-1,\ldots,n_{m}-1$ ).

We have already said that the properties of No. 20 do not extend, without modifications, to pseudo-polynomials of more than two variables. Consider, for example, the pseudo-polynomial

f(x,y,z)=\mathrm{A}(x,y)+\mathrm{B}(y,z)+\mathrm{C}(z,x)

of order $(0,0,0)$ , of three variables $x,y,z$ . If $f(x,y,z)$ is bounded the coefficients $A,B,C$ are not necessarily limited.

This results from the fact that these coefficients are not completely determined by the pseudo-polynomial considered. More precisely the pseudo-polynomial

[\mathrm{A}(x,y)+\phi(x)]+\mathrm{B}(y,z)+[\mathrm{C}(z,x)-\phi(x)]

Or $\phi(x)$ is any function of $x$ , is identical to $f(x,y,z)$ .
For a pseudo-polynomial of order ( $n_{i},n_{2},\ldots.,n_{m}$ ) we can write
(49) $\left[x_{1},x_{10},x_{11},\ldots,x_{1n_{1}}\left|x_{2},x_{20},x_{21},\ldots,x_{2n_{2}}\right|\ldots\left|x_{m},x_{m0},x_{m1},\ldots,x_{mn_{m}};f\right|=0\right.$ Or $x_{ji}$ are fixed values and $x_{1},x_{2},\ldots,x_{m}$ variables. From this relationship we can derive the value of $f\left(x_{1},x_{2},\ldots,x_{m}\right)$ . This formula gives as coefficients of the pseudo-polynomial linear expressions with respect to function of $1,2,\ldots$ , Or $m-1$ variables which are obtained by fixing, in the function $f\left(x_{1},x_{2},\ldots,x_{m}\right),m-1,m-2,\ldots$ or 1 of the variables. The coefficients of two identical pseudo-polynomials differ by pseudo-polynomials in $m-1$ variables $x_{1},x_{2},\ldots,x_{m}$ . When $m>2$ the coefficients of a pseudo-polynomial are determined up to expressions which contain arbitrary functions. Relation (49) shows us, however, that we have the following properties.

We can write the coefficients of a bounded pseudo-polynomial in such a way that these coefficients are bounded functions in their domain of existence.

We can write the coefficients of a continuous pseudo-polynomial in such a way that these coefficients are continuous functions in their domain of existence.

Let's say that a function $f\left(x_{1},x_{2},\ldots,x_{m}{}^{\prime}\right.$ is linearly measurable if it is measurable with respect to each of the variables $x_{1},x_{2},\ldots,x_{m}$ . For example, any measurable function $B$ is linearly measurable $B$ .

In order not to unnecessarily complicate our exposition, let us consider only linearly measurable pseudo-polynomials. We then have the following property:

The coefficients of a linearly measurable pseudo-polynomial can be written in such a way that these coefficients are linearly measurable functions in their domain of existence.

Conversely:
If the coefficients of a pseudo-polynomial are linearly measurable, this pseudo-polynomial is linearly measurable.

If the pseudo-polynomial is measurable B, we can write its coefficients so that they are measurable functions $B$ and vice versa.
24. - The properties of No. 21 are immediately generalized. It is sufficient here to state these properties.

If a difference of order ( $n_{1},n_{3},\ldots,n_{m}$ ) of the first kind of function $f\left(x_{1},x_{2},\ldots,x_{m}\right)$ is zero identically in R , we can construct a network which divides rationally and which is everywhere dense in R such that any partial divided difference of order $\left(n_{1},n_{2},\ldots,n_{m}\right)$ , taken on $\left(n_{1}+1\right)\left(n_{2}+1\right)\ldots\left(n_{m}+1\right)$ points chosen from the nodes of this network, or zero.

If a difference of order ( $n_{1},n_{2},\ldots,n_{m}$ ) of the first kind of the continuous function $f\left(x_{1},x_{2},\ldots,x_{m}\right)$ is identically zero, the partial divided difference of order ( $n_{1},n_{2},\ldots,n_{m}$ ) is identically zero in D . The function is therefore reduced in D to a pseudo-polynomial of order ( $n_{1}-1,n_{2},-1,\ldots,n_{n}-1$ ).

Let the functional equation of the same species be

\Delta_{h_{1}}^{n_{1}},h_{2},\ldots,n_{m},\ldots,h_{m}f\left(x_{1},x_{2},\ldots,x_{m}\right)=0

(50)

MA Marchaud's theorem extends to the case of $m$ variables,

Any bounded solution of equation (50) is a pseudo-polynomial of order ( $n_{1}-1,n_{2}-1,\ldots,n_{m}-1$ ) in R .

Let us demonstrate this by complete induction. We have seen that the property is true for $m=2$ . Let us demonstrate that if it is true for $m-1$ variables, it will also be true for $m$ variables. It suffices to demonstrate that if $f\left(x_{1},x_{2},\ldots,x_{m}\right)$ vanishes on an order network $\left(n_{1}-1,n_{2}-1,\ldots,n_{m}-1\right)$ , it is zero identically. Let us set
$g\left(x_{1},x_{2},\ldots,x_{m-1}\right)=\sum_{i=0}^{n_{m}}(-1)^{n_{m}-i}\binom{n_{m}}{i}f\left(x_{1},x_{2},\ldots,x_{m-1},x_{m}+ih_{m}\right)$ .
Giving to $x_{m}$ And $h_{m}$ fixed values, we have

\Delta_{h_{1}}^{h_{1}},h_{2},\ldots,h_{m},\ldots,h_{m-1}g\left(x_{1},x_{2},\ldots,x_{m-1}\right)=0

But, $g\left(x_{1},x_{2},\ldots,x_{m-1}\right)$ is a bounded and zero function on a network of order ( $n_{1}-1,n_{2}-1,\ldots,n_{m-1}-1$ ) therefore, by hypothesis, is identically zero. Now giving to $x_{1},x_{2},\ldots,x_{m-1}$ fixed values the relationship $g\left(x_{1},x_{2},\ldots,x_{m-1}\right)\equiv 0$ shows us that
$f\left(x_{1},x_{2},\ldots,x_{m}\right)$ is a polynomial of degree $n_{m}-1$ in $x_{m}$ . This polynomial being zero for $n_{m}$ values of $x_{m}$ is zero identically, which demonstrates the property.
25. - To be able to understand the form of the solutions to the equations that we will study in the following chapter, we must still specify a little the form of the pseudo-polynomials.

We will say that a polynomial in $y_{1},y_{v},\ldots,y_{r}$ is of degree $\left(s_{1},s_{2},\ldots,s_{r}\right)$ if it is of degree $s_{1}$ in $y_{1}$ , of degree $s_{2}$ in $y_{2},\ldots$ , of degree $s_{r}$ in $y_{r}$ .

Consider, in the field of functions at $m$ variables $x_{1}$ , $x_{2},\ldots,x_{m}$ , a polynomial $\mathrm{P}\left(x_{1},x_{2},\ldots,x_{r}\right)$ of degree $\left(s_{1},s_{2},\ldots,s_{r}\right)$ . We will say that $\mathrm{P}\left(x_{1},x_{2},\ldots,x_{r}\right)$ is any polynomial in $r$ variables if its coefficients are arbitrary functions pai with respect to the remaining variables $x_{r+1},x_{r+2},\ldots,x_{m}$ . This polynomial is therefore of the form

\mathrm{P}\left(x_{1},x_{2},\ldots,x_{r}\right)=\sum_{i_{1}=0}^{s_{1}}\sum_{i_{2}=0}^{s_{2}}\ldots\sum_{i_{r}=0}^{s_{r}}\mathrm{C}_{i_{1}i_{2}\ldots i_{r}}x_{1}^{i_{1}}x_{2}^{i_{2}}\ldots x_{r}^{i_{r}},

Or $\mathrm{C}_{i_{1}i_{2}}\ldots i_{r}$ are arbitrary functions of $r_{r+1},r_{r+2},\ldots,x_{m}$ .
A function which is a sum of any polynomials in $r$ variables will be called any pseudo-polynomial of the rth species. The structure of such a function is such that if it contains a term of the form

x_{j_{1}}^{l_{1}}x_{j_{2}}^{l_{2}}\ldots x_{j_{r}}^{l_{r}}\mathrm{\penalty 10000\ A}\left(x_{j_{r+1}},x_{j_{r+2}},\ldots,x_{j_{m}}\right)

where the coefficient A is an arbitrary function, it also contains all terms of the form

\begin{gathered}x_{j_{1}}^{l_{1}^{\prime}}x_{j_{2}}^{l_{2}^{\prime}}\ldots x_{j_{r}}^{l_{r}^{\prime}}\mathrm{A}\left(x_{j_{r+1}},x_{j_{r+2}},\ldots,x_{j_{m}}\right)\\ l_{1}^{\prime}\leqq l_{1},l_{2}^{\prime}\leqq l_{2},\ldots,l_{r}^{\prime}\leqq l_{r}\end{gathered}

In the following we will further specify by saying that these are any linearly measurable pseudo-polynomials of various kinds. We then imply that the coefficients are any linearly measurable functions.

CHAPTER IV

On some functional equations with several independent variables

26.
- —
  
  Now consider the equation at $m$ variables (of $m^{\text{ème }}$ species)

\Delta_{h_{1},h_{2},\ldots,h_{m}}^{\left(\alpha_{1i},\alpha_{2i},\ldots,\alpha_{mi}\right)}f\left(\dot{x}_{1},x_{2},\ldots,x_{m}\right)=0

(51)

We propose to search for the functions $f\left(x_{1},x_{2},\ldots,x_{m}\right)$ which verify equation (51) for all values $x_{1},x_{2},\ldots,x_{m}$ , $h_{1},h_{-},\ldots,h_{m}$ such as points ( $x_{1}+\alpha_{1i_{1}}h_{1},x_{2}+\alpha_{2i_{3}}h_{2},\ldots,x_{m}+\alpha_{mi_{m}}h_{m}$ ) are in the domain D. For simplicity, we can assume that $D$ be a rectangle $R$ , which does not restrict the generality of our results.

When it comes to the general equation (51), we make the assumptions indicated in No. 6 and, in particular, we assume that we have the inequalities (12). We generally assume that equation (51) is of order ( $k_{1},k_{2},\ldots,k_{m}$ ).

We also have the following properties:
Any function verifying equation (51) also verifies any consequent equation.

In particular:
Any function verifying equation (51) also verifies an equation of the form

\delta_{h_{1},h_{2},\ldots,h_{m}}^{\left(\alpha_{1i},\alpha_{2i},\ldots,\alpha_{mi}\right)}f\left(x_{1},x_{2},\ldots,x_{m}\right)=0

(52)

Any function verifying a reducible equation also verifies an equation of the form (50).

The first result demonstrated in No. 13 can be generalized. We will say that a function verifies a property around a network $\left(x_{10}\left|x_{20}\right|\ldots\mid x_{m0}\right)$ of order $(0,0,\ldots,0)$ , if it verifies this property in the domain formed by the bands $x_{i0}^{\prime}\leqq x_{i}\leqq x_{i0}^{\prime\prime},i=1$ ,
$2,\ldots,m$ , containing this network ( $a_{i}\leqq x_{i0}^{\prime}<x_{i0}^{\prime\prime}\leqq b_{i},x_{i0}^{\prime}\leqq x_{i0}\leqq x_{i0}^{\prime\prime}$ ). We then have the property:

Any solution to equation (51) that is bounded around a lattice of order ( $0,0,\ldots,0$ ), is bounded in R .

It is enough to give the demonstration for two variables $x$ And $y$ . So let equation (51) be with $m=2,n_{1}=m,n_{2}=n$ . We can assume $0=\alpha_{10}<\alpha_{11}<\ldots<\alpha_{1m},0=\alpha_{20}<\alpha_{21}<\ldots<\alpha_{2n}$ and that the solution $f(x,y)$ be bounded around the network $\left(b_{1}\mid b_{2}\right)$ , SO $|f(x,y)|<\mathrm{M}$ if $x$ is in the interval ( $b_{1}^{\prime},b_{1}$ ) Or $y$ in the meantime $\left(b_{2}^{\prime},b_{2}\right)$ . La démonstration se fait exactement comme au Nr. 13 dans le cas d’une seule variable. Nous pouvons encore supposser que $a_{00}\neq 0$ , autrement nous raisonnerions sur l’équation de la forme (52) à laquelle se réduit l’équation $\mathdollar(51)$ . Posant

\rho_{1}=\frac{\alpha_{11}}{\alpha_{1m}-\alpha_{11}},\quad\rho=\frac{\alpha_{21}}{\alpha_{2n}-\alpha_{21}},\quad\lambda=\frac{\sum_{i=0}^{m}\sum_{j=0}^{n}\left|a_{ij}\right|}{\left|\alpha_{j0}\right|}-1,

nous en déduisons

|f(x,y)|<\lambda\mathrm{M}\text{ pour }x>b_{1}^{\prime}-\rho_{1}\left(b_{1}-b_{1}^{\prime}\right)\text{ et }y>b_{2}^{\prime}-\rho_{2}\left(b_{2}-b_{2}^{\prime}\right).

Nous trouvons ensuite

		$\displaystyle\|f(x,y)\|<\lambda^{2}\mathrm{M}\text{ pour }x>b_{1}^{\prime}-\left[\left(1+\rho_{1}\right)^{2}-1\right]\left(b_{1}-b_{1}^{\prime}\right)\text{ et }y>b_{2}^{\prime}-\rho_{2}\left(b_{2}-b_{2}^{\prime}\right)$
		$\displaystyle\|f(x,y)\|<\lambda^{2}\mathrm{M}\text{ pour }x>b_{1}^{\prime}-\rho_{1}\left(b_{1}-b_{1}^{\prime}\right)\text{ et }y>b_{1}^{\prime}-\left[\left(1+\rho_{2}\right)^{2}-11\left(b_{2}-b_{2}^{\prime}\right).\right.$

Ces inégalités se déduisent toujours en écrivant l’équation (51) pour des valeurs convenables de $x,y,h_{1}$ et $h_{2}$ . De la même manière on obtient

|f(x,y)|<\lambda^{3}\mathbf{M}

pour $x>b^{\prime}{}_{1}-\left[\left(1+\rho_{1}\right)^{2}-1\right]\left(b_{1}-b^{\prime}{}_{1}\right)$ et $y>b^{\prime}{}_{2}-\left[\left(1+\rho_{2}\right)^{2}-1\right]\left(b_{2}-b^{\prime}{}_{2}\right)$ .
En répétant le procédé, on déduit que

|f(x,y)|<\lambda^{2s-1}M

pour $x>b_{1}^{\prime}-\left[\left(1+\rho_{1}\right)^{s}-1\right]\left(b_{1}-b_{1}^{\prime}\right)$ et $y>b_{2}^{\prime}-\left[\left(1+\rho_{2}\right)^{s}-1\right]\left(b_{2}-b_{2}^{\prime}\right)$ , donc

|f(x,y)|<\lambda^{2s-1}\mathbf{M}\text{ dans }\mathrm{R}

si $\left(1+p_{1}\right)^{s}>\frac{b_{1}-a_{1}}{b_{1}-b_{1}^{\prime}},\left(1+p_{2}\right)^{s}>\frac{b_{2}-a_{2}}{b_{2}-b_{2}^{\prime}}$ .
La démonstration se fait exactement de la même manière pour $m$ quelconque.

Nous en déduisons la propriété suivante :
Si deux solutions de l’équation (51) coüncident autour d’un réseau d’ordre $(0,0,\ldots,0$ , elle coüncident partout dans R.

En particulier :
Toute solution de l’équation (51) qui se réduit à un pseudopolynome autour d’un réseau d’ordre ( $0,0,\ldots,0$ ), est un pseudopolynome dans R.
27. - Occupons-nous maintenant de l’équation (52), supposée d’ordre ( $n_{1},n_{2},\ldots,n_{m}$ ).

Démontrons le théoreme suivant :
Toute solution linéairement mesurable de l’équation (52) est un pseudo-polynome d’ordre ( $n_{1}-1,n_{2}-1,\ldots,n_{m}-1$ ).

Nous démontrons ce théorème par induction complète. Il euffit évidemment de démontrer que si cette solution s’annule sur un réseau d’ordre ( $n_{1}-1,n_{2}-1,\ldots,n_{m}-1$ ), elle est nulle identiquement. Posons

g\left(x_{1}\right)=\delta_{h_{2}}^{\left(a_{2i}\right)}\delta_{h_{3}}^{\left(a_{3i}\right)}\ldots\delta_{h_{mi}}^{\left(a_{mi}\right)}f\left(x_{1},x_{2},\ldots,x_{m}\right)

Nous avons $\delta_{h_{1}}^{\left(a_{1i}\right)}g\left(x_{1}\right)=0$ . Mais, les $x_{2},x_{3},\ldots x_{m},h_{2},h_{3},\ldots,h_{m}$ étant donnés, $g\left(x_{1}\right)$ est une fonction mesurable de $x_{1}$ , donc elle se réduit à un polynome de degré $n_{1}-1$ qui, étant nul pour $n_{1}$ valeurs de $x_{1}$ , est nul identiquement. La fonction $f\left(x_{1},x_{2},\ldots,x_{m}\right)$ vérifie donc l’équation

\delta_{h_{2}}^{\left(u_{2i}\right)}\delta_{h_{3}}^{\left(\alpha_{3i}\right)}\ldots\delta_{h_{mi}}^{\left(\alpha_{mi}\right)}f\left(x_{1},x_{2},\ldots,x_{m}\right)=0

pour toute valeur donnée de $x_{1}$ . Le théorème étant supposé vrai pour $m-1$ variables, il en résulte qu’il est aussi vrai pour $m$ variables.

Nous pouvons énoncer maintenant le théorème suivant :
La solution linéairement mesurable générale de l’équation (51) est la même que sa solution générale dans le champ des pseudo-polynomes linéairement mesurables.

En particulier :
La solution mesurable B générale de l’équation (51) est la même que sa solution générale dans le champ des pseudo-polynomes mesurables B.
28. - II reste à trouver maintenant la solution générale de l’équation (51) dans le champ des pseudo-polynomes.

Supposons que l’équation (51) ait un ordre $r^{uple}\left[k_{1}^{\prime},k_{2}^{\prime},\ldots,k_{r}^{\prime}\right]_{1,2,\ldots,r}$ ( $1\leqq r\leqq m$ ) et cherchons alors une solution de la forme

x_{1}^{l_{1}}x_{2}^{l_{2}}\ldots x_{r}^{l_{r}}\mathbf{A}\left(x_{r+1},x_{r+2},\ldots,x_{m}\right)

(53)

avec A fonction arbitraire.
On trouve que les conditions nécessaires et suffisantes pour qu’il en soit ainsi sont précisément les égalités (19) pour $v_{s}=0,1,\ldots,l_{s}$ , $s=1,2,\ldots,r$ . Nous avons donc la propriété suivante :

Si l’équation (51) a l’ordre ruple $\left[k_{1}^{\prime},k_{2}^{\prime},\ldots,k_{r}^{\prime}\right]_{1,2,\ldots,r}$ elle est vérifiée par un polynome quelconque de degré ( $k_{1}^{\prime},k_{2}^{\prime},\ldots,k_{r}^{\prime}$ ) en $x_{1},x_{2},\ldots,x_{r}$ .

En particulier :
Tout pseudo-polynome d’ordre ( $k_{1}-1,k_{2}-1,\ldots,k_{m}-1$ ) vérifie l’équation (51), supposée d’ordre ( $k_{1},k_{2},\ldots,k_{m}$ ).

Pour trouver la solution générale dans le champ des pseudopolynomes linéairement mesurables il suffit de chercher les solutions de la forme
(54) $\mathrm{G}\left(x_{1},x_{2},\ldots,x_{m}\right)=\sum_{j=1}^{m}\sum_{i=k_{j}}^{n_{j}^{\prime}}x_{j}^{i}\mathrm{\penalty 10000\ A}_{ji}\left(x_{1},x_{2},\ldots,x_{j-1},x_{j+1},\ldots,x_{m}\right)$ .

We know that we can assume the coefficients to be linearly measurable. We have

	$\displaystyle\Delta_{h_{2}}^{n_{2}^{\prime}+1,h_{3},\ldots,n_{m}^{\prime}+1,\ldots,n^{\prime}m+1}\mathrm{G}$	$\displaystyle\left(x_{1},x_{2},\ldots,x_{m}\right)=$
		$\displaystyle=\sum_{i=k_{1}}^{n^{\prime}}x_{1}^{i}\Delta_{h_{2},h_{3},\ldots,h_{m}}^{n_{2}^{\prime}+1,n_{3}^{\prime}+1,\ldots,n_{m}^{\prime}+1}\mathrm{\penalty 10000\ A}_{1i}\left(x_{2},\ldots,x_{m}\right)$

Or $\Delta_{h_{2},h_{\mathrm{a}},\ldots,h_{m}}^{n_{\mathrm{a}}+1,n_{\mathrm{a}}^{\prime}+1,\ldots,n^{\prime}m+1}$ is the difference operation defined in No. 6 and executed on the variables $x_{2},x_{3},\ldots,x_{m}$ . Now, the difference operation and operation (11) are intermutable. By writing that $\mathrm{F}\left(x_{1},x_{2},\ldots,x_{m}\right)$ check the equation, we therefore find

\begin{gathered}x_{1}^{n_{1}^{\prime}-k_{1}}\sum_{i_{2}=0}^{n_{2}}\sum_{i_{3}=0}^{n_{3}}\ldots\sum_{i_{m}=0}^{n_{m}}\gamma_{i_{2}i_{3}\ldots i_{m}}^{\left(k_{1}\right)}\Delta_{h_{2},h_{3},\ldots,h_{m}}^{n_{9}^{\prime}+1,n_{3}^{\prime}+1,\ldots,n_{m}^{\prime}+1}\mathrm{\penalty 10000\ A}_{1n_{1}^{\prime}}\left(x_{2},x_{3},\ldots,x_{m}\right)+\\ +\cdots=0\end{gathered}

the unwritten terms forming a polynomial of degree $n_{1}^{\prime}-k_{1}-1$ in $x_{1}$ .

We must therefore have

\text{ (55) }\sum_{i_{2}=0}^{n_{2}}\sum_{i_{3}=0}^{n_{3}}\ldots\sum_{i_{m}=0}^{n_{m}}\gamma_{i_{2}i_{3}}^{\left(k_{1}\right)}\ldots i_{m}\Delta_{h_{2}}^{n_{2}^{\prime}+1,h_{3},\ldots,n_{m}^{\prime}+1,\ldots,n^{\prime}{}_{m}}{}^{1}\mathrm{\penalty 10000\ A}_{1n_{1}^{\prime}}\left(x_{2},x_{3},\ldots,x_{m}\right)=0.

This equation is of the form (51), in $m-1$ variables and, as a result of the definition of the number $k_{1}$ , its coefficients are not all zero. It follows that $\mathbf{A}_{1n_{1}^{\prime}}$ is a pseudo-polynomial. In $\mathrm{G}\left(x_{1},x_{2},\ldots,x_{m}\right)$ we can fit the term in $x_{1}^{n^{\prime}}\mathrm{A}_{1n}$ in terms $x_{j}^{i}\mathrm{\penalty 10000\ A}_{ji},j>1$ . The solution sought is therefore of the form (54) where the $n_{j}^{\prime}$ are, in general, changed but $n_{1}^{\prime}$ is replaced by $n_{1}^{\prime}-1$ . Repeating the process, we demonstrate that the $\mathrm{A}_{1i}$ are all pseudo-polynomials. We demonstrate in the same way that the $\mathbf{A}_{2i},\mathbf{A}_{3i},\ldots,\mathbf{A}_{ni}$ are all pseudo-polynomials.

Finally, the general linearly measurable solution of equation (51) is of the form

\mathrm{G}\left(x_{1},x_{2},\ldots,x_{m}\right)=\mathrm{P}_{1}+\mathrm{P}_{2}+\cdots+\mathrm{P}_{m},

(56)

Or $\mathrm{P}_{1}$ is any pseudo-polynomial of order ( $k_{1}-1,k_{2}-1,\ldots,k_{m}-1$ ) and, in general, $\mathrm{P}_{r}$ is a sum of functions of the form (53) containing arbitrary functions of $m-r$ variables. In particular, $\mathrm{P}_{m}$ is a polynomial in $x_{1},x_{2},\ldots,x_{m}$ .
29. - We can now demonstrate the following general theorem:

The general linearly measurable solution of equation (51) is of the form (56) where $\mathrm{P}_{r}$ is any pseudo-polynomial of the rth species. The pseudo-polynomial $\mathrm{P}_{r}$ is a sum of any polynomials with r variables, to each ruple order corresponding such a polynomial.

First of all the fact that $\mathbf{P}_{\mathbf{r}}$ is a pseudo-polynomial of the rth species does not yet result from what precedes. This result is easily demonstrated by complete induction. Indeed, by assuming true the property for $m-1$ variables, the previous results show us that it is also true for $m$ variables. The induction is complete since the property is obviously true for $m=1$ .

It should be noted that the proof of the theorem is not yet complete. Indeed, it is still necessary to demonstrate that $\mathbf{P}_{r}$ can actually be put in the form of a sum of functions
of the form

a_{j_{1}}^{l_{1}}x_{j_{2}}^{l_{2}}\ldots x_{j_{r}}^{l_{r}}\mathbf{A}\left(x_{j_{r+1}},x_{j_{r+2}},\ldots,x_{j_{m}}\right)\text{ (A fonction arbitraire) }

(57)

which are solutions of equation (51). This fact does not follow immediately from the above because of equations (55) where the operation $\Delta$ has the effect of raising too high the order of the equations which the functions must satisfy $\mathbf{A}_{ji}$ ,

We demonstrate this property step by step for $\mathrm{P}_{1}$ , $\mathbf{P}_{2},\ldots$ , And $\mathbf{P}_{m}$ . The property is obvious to $\mathbf{P}_{1}$ . It will be enough to indicate the progress of the demonstration by doing it for $P_{2}$ . The term $P_{2}$ can be split in two, $P_{2}=P^{\prime}{}_{2}+P_{2}^{\prime\prime}$ Or $P_{2}$ contains the terms $(57)(r=2)$ in which $j_{1}$ And $j_{2}$ are 1 and 2 and $\mathrm{P}_{2}^{\prime\prime}$ the other terms of $\mathrm{P}_{2}$ . We can find integers $r_{3},r_{4},\ldots,r_{m}$ large enough such that

\begin{gathered}\Delta_{h_{3}}^{r_{3}},r_{4},\ldots,r_{m},\ldots,h_{m}\mathrm{P}_{2}=\Delta_{h_{3}}^{r_{3}},r_{4},\ldots,r_{m},\ldots,h_{m}\mathrm{P}_{2}^{\prime}\\ \Delta_{h_{9}}^{r_{3}},r_{4},\ldots,h_{4},\ldots,h_{m}\mathrm{G}\left(x_{1},x_{2},\ldots,x_{m}\right)=\Delta_{h_{3}}^{r_{3}},r_{4},\ldots,r_{4},\ldots,h_{m}\mathrm{P}_{1}+\Delta_{h_{3}}^{r_{3}},r_{4},\ldots,r_{m},\ldots,h_{m}\mathrm{P}_{2}^{\prime}\end{gathered}

the operation $\Delta$ relating to variables $x_{3},x_{4},\ldots,x_{m}$ .
Now we just have to write that

\Delta_{h_{1},h_{2},\ldots,h_{m}}^{\left(a_{1i},a_{2i},\ldots,a_{mi}\right)}\Delta_{h_{3},h_{4},\ldots,h_{m}}^{r_{9}},r_{4},\ldots,r_{m}\mathrm{P}_{2}^{\prime}\left(x_{1},x_{l},\ldots,x_{m}\right)=0

is verified identically in $x_{1},x_{2},h_{1},h_{2}$ . Either

x_{1}^{l_{1}}x_{2}^{l_{2}}\mathrm{\penalty 10000\ A}\left(x_{3},x_{4},\ldots,x_{m}\right)

(58)

one of the highest degree terms in $x_{1},x_{2}$ of $\mathrm{P}_{2}^{\prime},l_{1}\equiv k_{1}$ , $l_{2}\equiv k_{2}$ . We deduce that the function $\mathrm{A}\left(x_{3},x_{4},\ldots,x_{m}\right)$ must verify the equations

\begin{gathered}\sum_{i_{3}=0}^{n_{3}}\sum_{i_{4}=0}^{n_{4}}\ldots\sum_{i_{m}=0}^{n_{m}}\gamma_{i_{0}i_{4}\ldots i_{m}}^{\left(\nu_{1},\nu_{2}\right)}\Delta_{h_{3},h_{4},\ldots,r_{m}}^{r_{3}},r_{4},\ldots,r_{m}A\left(x_{3},x_{4},\ldots,x_{m}\right)=0\\ \nu_{1}=k_{1},k_{1}+1,\ldots,l_{1},v_{2}=k_{2},k_{2}+1,\ldots,l_{2}\end{gathered}

If (58) is not a solution of (51) at least one of these equations has coefficients not all zero, therefore $\mathbf{A}\left(x_{3},x_{4},\ldots,x_{m}\right)$ is a pseudo-polynomial. In this case the term (58) of $\mathbf{P}_{2}$ can be put in one of the $\mathrm{P}_{r},r>2$ . By repeating this process a sufficient number of times and making the same demonstration for the
other terms of $P_{2}$ , we arrive at the demonstration of the property for $\mathrm{P}_{2}$ . We then demonstrate, in the same way, the property for $\mathrm{P}_{3},\mathrm{P}_{4},\ldots,\mathrm{P}_{m}$ . For the latter, we easily verify that any polynomial verifying the equation is a sum of solutions of the form $\mathrm{A}x_{1}^{l_{1}}x_{2}^{l_{2}}\ldots x_{m}^{l_{m}}$ , A being an arbitrary constant.

The general theorem holds, in particular, assuming the solution is measurable $B$ . In this case in the general solution we can assume the arbitrary measurable functions $B$ .
30. - Let us put the preceding results into a clearer form. It follows, in fact, from the preceding analysis that we can state our fundamental result in the following form:

The general linearly measurable solution of equation (51) is of the form

\sum_{j=1}^{m}\sum_{i=0}^{n_{j}^{\prime}}x_{j}^{i}\mathrm{\penalty 10000\ A}_{ji}\left(x_{1},x_{2},\ldots,x_{j-1},x_{j+1},\ldots,x_{m}\right)

$\mathrm{A}_{ji}$ being the most general linearly measurable function satisfying equations
(59)

\begin{gathered}\sum_{i_{1}=0}^{n_{1}}\ldots\sum_{i_{j-1}=0}^{n_{j-1}}\sum_{i_{j+1}=0}^{n_{j+1}}\sum_{i_{m}=0}^{n_{m}}\gamma_{i_{1}\ldots i_{j-1}i_{j}+1\ldots i_{m}}^{(\nu)}\mathbf{A}_{ji}\left(\ldots,x_{r}+\alpha_{ri_{r}}h_{r},\ldots\right)=0\\ \nu=0,1,\ldots,n_{j}\end{gathered}

relative to variables $x_{1},x_{2},\ldots,x_{j-1},x_{j+1},\ldots,x_{m}$ .
Let us look for example for the conditions so that in the general solution (56) the terms $\mathbf{P}_{\mathbf{1}},\mathbf{P}_{\mathbf{2}},\ldots,\mathbf{P}_{r-\mathbf{1}}$ disappear.

The definitions and results of No. 7 show us that:
The necessary and sufficient condition for the general linearly measurable solution of equation (51) to contain only arbitrary functions of at most $r$ variables is that all the ( $m-r-2$ )th derived equations are of order ( $0,0,\ldots,0$ ).

In particular, for the general solution to contain only arbitrary functions of a single variable, it is necessary and sufficient that all the ( $m-3$ ) th derived equations are of order ( $0,0,0$ ).

Similarly:
The necessary and sufficient condition for the general linearly measurable solution of equation (51) to be a polynomial
is that all the ( $m-2$ )th derived equations are of order $(0,0)$ .

For example if $m=2$ , the necessary and sufficient condition for the general linearly measurable solution to be a polynomial is that the equation is of order ( 0,0 ). We have already partially established this result in our previous work [11].

We also see that:
The necessary and sufficient condition for the general linearly measurable solution of equation (51) of order ( $k_{1},k_{2},\ldots,k_{m}$ ), or a pseudo-polynomial of order ( $k_{1}-1,k_{2}-1,\ldots,k_{m}-1$ ) is that the equation (or the expression of its first member) has no double, triple, …, multiple order.

Equation (52) enjoys, in particular, this property.
Consider $\mathrm{N}=\left(n_{1}+1\right)\left(n_{\iota}+1\right)\ldots\left(n_{m}+1\right)$ points $\mathrm{M}_{i}\left(x_{1i},x_{2i},\ldots,x_{mi}\right)i=1,2,\ldots,\mathrm{\penalty 10000\ N}$ in hyperspace. We can form the determinant $\mathrm{V}_{n_{1},n_{2},\ldots,n_{m}}\left(\mathbf{M}_{1},\mathbf{M}_{2},\ldots,\mathbf{M}_{\mathrm{N}}\right)$ has $\mathbf{N}$ rows and columns whose general line is formed by the elements $x_{1i}^{v_{1}}x_{2i}^{v_{2}}\ldots x_{mi}^{v_{m}},v_{j}=0$ , $1,\ldots,n_{j},j=1,2,\ldots,m$ . We deduce the determinant $\mathrm{U}_{n_{1},n_{2},\ldots,n_{m}}\left(\mathrm{M}_{1},\mathrm{M}_{2},\ldots,\mathrm{M}_{\mathrm{N}};f\right)$ , by replacing the elements $x_{1i}^{n_{1}}x_{2i}^{n_{2}}\ldots x_{mi}^{n_{m}}$ by $f\left(x_{1i},x_{2i},\ldots,x_{mi}\right)$ respectively. The quotient

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

is the divided difference of order ( $n_{1},n_{2},\ldots,n_{m}$ ) of the function $f\left(x_{1},x_{2},\ldots,x_{m}\right)$ on the points $\mathrm{M}_{i}$ . This definition only makes sense, of course, if the determinant of the denominator is $\neq 0$ . For $m=2$ we have already given this definition [10].

If we pose $x_{ji}=x_{j}+a_{ji}h_{j},i=1,2,\ldots,\mathrm{\penalty 10000\ N}-1,x_{j\mathrm{\penalty 10000\ N}}=x_{j}$ , $j=1,2,\ldots,m$ , the equation

\left[\mathbf{M}_{1},\mathbf{M}_{2},\ldots,\mathbf{M}_{\mathrm{N}};f\right]_{n_{1},n_{2},\ldots,n_{m}}=0

is of the form (51). This equation is, in general, of order $(0,0,\ldots,0)$ and is verified by any polynomial of degree $\left(n_{1},n_{2},\ldots,n_{m}\right)$ which does not contain a term in $x_{1}^{n_{1}}x_{2}^{n_{2}}\ldots x_{m}^{n_{m}}$ . Let, in particular, be the equation

\begin{gathered}{\left[x_{1},x_{1}+\alpha_{11}h_{1},\ldots,x_{1}+\alpha_{1n_{1}}h_{1}\left|x_{2},x_{2}+\alpha_{21}h_{2},\ldots,x_{2}+\alpha_{2n_{2}}h_{2}\right|\ldots\right.}\\ \left.\ldots\mid x_{m},x_{m}+\alpha_{m}h_{m},\ldots,x_{m}+\alpha_{mi_{m}}h_{m};f\right]=0\end{gathered}

This equation is of order ( $n_{1},n_{-},\ldots,n_{m}$ ) and its
general linearly measurable solution is any polynomial of order $\left(i_{1}-1,n_{2}-1,\ldots,n_{m}-1\right)$ .
31. - We have always assumed that functions are linearly measurable. Let us consider, for simplicity, the case of two variables and let the equation

\delta_{h_{1},h_{2}}^{\left(a_{1i},a_{2i}\right)}f(x,y)=0,\quad\left(n_{1}=m,n_{2}=n\right)

(60)

Now consider a measurable solution $f(x,y)$ of equation (60). Let $\mathrm{E}_{x}$ the whole, of measurement $b_{1}-a_{1}$ , values $x$ for which $f(x,y)$ is a measurable function of $y$ And $\mathrm{E}_{y}$ the whole, of measurement $b_{4}-a_{2}$ , values $y$ for which $f(x,y)$ is a measurable function of $x$ . Either $\mathrm{P}(x,y)$ the pseudo-polynomial of order ( $m-1,n-1$ ) which takes the same values as $f(x,y)$ on the network ( $x_{0},x_{1},\ldots,x_{m-1}\mid y_{0},y_{1},\ldots,y_{n-1}$ ), Or $x_{i}\in\mathrm{E}_{x},y_{j}\in\mathrm{E}_{y}$ and let's ask $f_{1}(x,y)=f(x,y)-\mathrm{P}(x,y)$ . The function $f_{1}(x,y)$ verifies the equation, cancels on the network ( $x_{0},x_{1},\ldots,x_{m-1}\mid y_{0},y_{1},\ldots,y_{n-1}$ ), is measurable relative to $y$ For $x\in\mathrm{E}_{x}$ and is measurable relative to $x$ For $y\subset\mathrm{E}_{y}$ . If we pose $g(x)=\delta_{h_{\mathrm{a}}}^{\left(a_{2i}\right)}f_{1}(x,y)$ , the function $g(x)$ is measurable for $y+\alpha_{2i}h_{2}=\mathrm{E}_{y}$ and checks the equation $\delta_{h_{1}}^{\left(\alpha_{1i}\right)}g(x)=0$ . For $x\in\mathrm{E}_{x}$ the function $f_{1}(x,y)$ is measurable relative to $y$ and checks the equation $\delta_{h_{2}}^{\left(a_{2i}\right)}f_{1}(x,y)=0$ on $\mathrm{E}_{y}.f_{1}(x,y)$ is therefore a polynomial of degree $n-1$ in $y$ who, being zero for $y_{j}\subset\mathrm{E}_{y}$ , is zero identically on $\mathrm{E}_{y}(\mathrm{Nr}.17)$ . It follows that $f_{1}(x,y)=0$ if $x\in\mathrm{E}_{x},y=\mathrm{E}_{y}$ , SO

Any measurable solution of equation (60) reduces to a pseudo-polynomial, except perhaps on a set & formed by a set of zero measure of lines parallel to the Ox axis and by a set of zero measure of lines parallel to the axis $\mathrm{O}y$ .

We can easily construct an example which shows that the general theorem stated above (No. 29) cannot be extended to simply measurable functions. Consider the rectangle $\mathrm{R}[-1<x<+1,-1<y<+1]$ . Either $\chi(x)$ a discontinuous solution of the Cauchy equation $f(x+y)=f(x)+f(y)$ . We can choose this solution so that we have $\chi(0)=0$ And $\chi(x)=0$ on an everywhere dense set [7]. It suffices to take a solution such that $\chi(0)=0,\chi(1)=0$ , Let us then consider the function
$f(x,y)$ defined as follows

\begin{array}[]{ll}f(x,0)=\chi(x)&-1<x<+1\\ f(0,y)=\chi(y)&-1<y<+1\\ f(x,y)=0;&x\neq 0,y\neq 0\end{array}

This function verifies the equation, of the sort ( 60 ),

\Delta_{h_{1},h_{\mathbf{a}}}^{m_{1}n}f(x,y)=0,\quad m\geq 2,n\geq 2

but does not reduce to a pseudo-polynomial (otherwise the function should be identically zero).

There are, however, cases where it can be said that any measurable solution to equation (51) is a pseudo-polynomial. Or again $m=2,n_{1}=m,n_{2}=n$ . Suppose, for example, that in this equation $a_{i0}=0,i=1,2,\ldots,m,a_{0j}=0,j=1,2,\ldots,n$ . Either $f(x,y)$ a measurable solution of this equation and consider the set $\varepsilon$ corresponding. There exists a pseudo-polynomial $\mathrm{P}(x,y)$ such as $f(x,y)=\mathrm{P}(x,y)$ if the point $(x,y)$ does not belong to $\mathcal{E}$ . But, whatever $x$ And $y$ can find one $h_{1}$ and one $h_{2}$ such as points $\left(x+\alpha_{1i}h_{1},y+\alpha_{2j}h_{2}\right).i=1,2,\ldots,m,j=1,2,\ldots,n$ do not belong to $\mathscr{E}$ . It immediately follows that we have $f(x,y)=\mathrm{P}(x,y)$ everywhere.

Reducibility can also often be used to recognize whether an equation enjoys the previous property.
32. - As in the case of a single variable, we can consider reducible equations.

Any bounded solution to a reducible equation is a pseudopolynomial.

In general, one can even assert that the most general bounded solution of a reducible equation is the same as its solution in the field of pseudo-polynomials. This is surely so if equations (55), (59) are reducible.

This case certainly occurs for the equation

\left.\Delta_{h_{1}}^{\left(n_{1}\right.},n_{2},\ldots,n_{m},\ldots,h_{m}\right)f\left(x_{1},x_{2},\ldots,x_{m}\right)=0

(61)

which we have already examined for $m=2$ [11].
The general bounded solution of equation (61) is of the form (56), where all arbitrary function coefficients are any bounded functions.

For the general bounded solution of equation (61) to be a polynomial of degree $n_{1}-1$ in $x_{1}$ , of degree $n_{2}-1$ in $x_{2},\ldots$ , of degree $n_{m}-1$ in $x_{m}$ , it is necessary and sufficient that the characteristic polynomial of the first type be of the form
(62) $\quad\mathrm{F}^{*}\left(x_{1},x_{2},\ldots,x_{m}\right)=\Sigma c_{i_{1}i_{2}\ldots i_{m}}\left(1-x_{1}\right)^{i_{1}}\left(x-x_{2}\right)^{i_{2}}\ldots\left(1-x_{m}\right)^{i_{rn}}$ ,
where the summation is extended to the values $i_{j}=0,1,\ldots,n_{j},j=1,2,\ldots,m$ , the values for which we have both $i_{1}<n_{1},i_{2}<n_{2},\ldots,i_{m}<n_{m}$ being excluded. It is also necessary that the constants $c_{n_{1}00\ldots 0},c_{0n_{2}0\ldots 0}$ , …, $c_{00\ldots 0n}$ are all different from zero. Simple special cases are

\mathrm{F}^{*}\left(x_{1},x_{2},\ldots,x_{m}\right)=\sum_{i=1}^{m}\left(1-x_{i}\right)^{n_{i}}

$F^{*}\left(x_{1},x_{2},\ldots,x_{m}\right)=\sum_{i=0}^{m}\left(1-x_{i}\right)^{n_{i}}\left(1+x_{1}\right)^{n_{1}\ldots\left(1+x_{i-1}\right)^{n_{i-1}}\left(1+x_{i+1}\right)^{n_{i+1}}\ldots\left(1+x_{m}\right)^{n_{m}}}$ .
For the general bounded solution of equation (61) to be any polynomial of degree $n-1$ it is necessary and sufficient that the characteristic polynomial of the first type be of the form (62), where the summation is extended to the values $i_{j}=0,1,\ldots,n_{j},j=1$ , $2,\ldots,m$ for which $i_{1}+i_{2}+\ldots+i_{m}\geqq n$ . It is also necessary that all constants $c_{i_{1}i_{2}\ldots i_{m}}$ , for which $i_{1}+i_{2}+\ldots+i_{m}=n$ , are different from zero. Simple special cases are

\mathrm{F}^{*}\left(x_{1},x_{2},\ldots,x_{m}\right)=\left(m-x_{1}-x_{2}-\cdots-x_{m}\right)^{n}

$\mathrm{F}^{*}\left(x_{1},x_{2},\ldots,x_{m}\right)=\left(x_{1}+x_{2}+\cdots+x_{\frac{m}{2}}-x_{\frac{m}{2}+1}-x_{\frac{m}{2}+2}-\cdots-x_{m}\right)^{n}$ if $m$ is even

\mathrm{F}^{*}\left(x_{1},x_{2},\ldots,x_{m}\right)=\Sigma\left(1-x_{1}\right)^{i_{1}}\left(1-x_{2}\right)^{i_{2}}\ldots\left(1-x_{m}\right)^{i_{m}}\ldots\text{ etc. }

We can also taciturnly see that if

F^{*}\left(x_{1},x_{2},\ldots,x_{m}\right)=\sum\left(1-x_{j_{1}}\right)^{n_{j_{1}}}\left(1-x_{j_{2}}\right)^{n_{j_{1}}}\ldots\left(1-x_{j_{r}}\right)^{n_{j_{r}}},

the summation being extended to all combinations $j_{1},j_{2},\ldots,j_{r}r$ has $r$ numbers $1,2,\ldots,m$ , the general solution contains only arbitrary functions of at most $r-1$ variables.

CHAPTER V

On some functional properties characterizing polynomials of two variables

33.
- —
  
  Consider the pseudo-polynomial of order ( $m,n$ ),

f(x,y)=\sum_{i=0}^{m}x^{i}A_{i}(y)+\sum_{j=0}^{n}y^{j}B_{j}(x)

defined in the domain $\mathbf{D}$ .
Let's do the change of variables

x=\alpha x^{\prime}+\beta y^{\prime}\quad y=\gamma x^{\prime}+\delta y^{\prime},

(63)

the function $f(x,y)$ then becomes $f_{1}\left(x^{\prime},y^{\prime}\right)$ .
Let us now try to determine the coefficients $\mathrm{A},\mathrm{B}$ of the pseudo-polynomial so that $f_{1}\left(x^{\prime},y^{\prime}\right)$ is still a pseudopolynomial in $x^{\prime}$ And $y^{\prime}$ . If so, we can find two positive integers $m^{\prime},n^{\prime}$ such that we have

\Delta_{\beta h,-\alpha h}^{m^{\prime},n^{\prime}}f_{1}\left(x^{\prime},y^{\prime}\right)=0

identically in $x^{\prime},y^{\prime},h$ . Taking into account (63) and applying the operation $\Delta_{\alpha\beta h}^{m+1}$ compared to $x$ , we find

\left.\left.\sum_{j=0}^{n}\sum_{r=0}^{m^{\prime}}\sum_{s=0}^{n^{\prime}}(-1)^{m^{\prime}+n^{\prime}-r-s}\binom{m^{\prime}}{r}\binom{n^{\prime}}{s}\right\rvert\,y+(r\beta\gamma-s\alpha\delta)h\right]^{j}\Delta_{a\beta h}^{m+1}B_{j}[x+(r-s)\alpha\beta h\mid=0

identically in $x,y,h$ . It is therefore necessary, in particular, that we have

\sum_{r=0}^{m^{\prime}}\sum_{s=0}^{n^{\prime}}(-1)^{m^{\prime}+n^{\prime}-r-s}\binom{m^{\prime}}{r}\binom{n^{\prime}}{s}\Delta_{\alpha\beta h}^{m+1}\mathrm{\penalty 10000\ B}_{n}[x+(r-s)\alpha\beta h]=0

or, after slight modifications,

\Delta_{a\beta h}^{m^{\prime}+n^{\prime}+m+1}\mathrm{\penalty 10000\ B}_{n}(x)=0.

If we are in the case where we can say that the general solution of this equation is a polynomial, we see that $f(x,y)$ is of the form

f(x,y)=\sum_{i=0}^{m_{1}}x^{i}A_{i}(y)+\sum_{j=0}^{n-1}y^{j}B_{j}(x)\quad\left(m_{\mathbf{1}}=m+m^{\prime}+n^{\prime}\right)

Repeating the process, we see that
and, in general,

\Delta_{\alpha\beta h}^{m^{\prime}+n^{\prime}+m_{1}+1}\mathrm{\penalty 10000\ B}_{n-1}(x)=0

\Delta_{\alpha\beta h}^{m+(n-j+1)\left(m^{\prime}+n^{\prime}\right)}\mathrm{B}_{j}(x)=0,\quad j=0,1,\ldots,n

(64)

under the assumption that the general solutions of the equations for $\mathrm{B}_{j+1}(x),\mathrm{B}_{j+2}(x),\ldots,\mathrm{B}_{n}(x)$ are polynomials. Under similar assumptions we find that

\Delta_{\gamma\delta h}^{n+(m-i+1)\left(m^{\prime}+n^{\prime}\right)}A_{i}(y)=0,\quad i=0,1,\ldots,m.

(65)

To simplify the language we will say that a transformation (63) defines a direction T. This transformation means that we ^have taken a new system of axes $\mathrm{O}x^{\prime}y^{\prime}$ . In particular, the initial Oxy system is the direction $T_{0}$ We can assume, without restricting generality, that the unit of length never changes and we can then assume, which we always will, that $x^{2}+\gamma^{2}=\beta^{2}+\delta^{2}=1$ . In this way the system $\mathrm{O}x^{\prime}y^{\prime}$ is completely determined by the coefficients $\alpha,\beta,\gamma,\delta$ of the transformation (63). We will say that two directions $\mathrm{T}_{1},\mathrm{\penalty 10000\ T}_{2}$ are completely distinct if the lines carrying the four axes are distinct. For the directions T and $\mathrm{T}_{0}$ are completely distinct it is necessary and sufficient that we have

\alpha\neq 0,\beta\neq 0,\gamma\neq 0,\delta\neq 0,\alpha\delta-\beta\gamma\neq 0.

If we notice that equations (64), (65) are reducible, we can state the following theorem:

The necessary and sufficient condition for a bounded function, or for a measurable function, to be a pseudo-polynomial in two completely distinct directions is that this function reduces to a polynomial.

The condition is obviously sufficient. We have shown that it is also necessary for $\mathrm{T}_{0}$ and T which does not restrict the generality. We have taken into account the properties of chap. III. The theorem is true for the domain D. In the demonstration the variables $x,y,h$ vary in such a way that we do not leave the domain D. We can also reason step by step, by decomposing the domain D into suitable partial domains. The conclusions are perfectly justified because of the extension property enjoyed by pseudopolynomials (No. 20).

It can easily be seen that if the two directions are not completely distinct the property is no longer true.
34. - Let us now determine the general form of the polynomials which are pseudo-polynomials of given order ( $m,n$ ) in the initial direction $\mathrm{T}_{0}$ and given order ( $m^{\prime},n^{\prime}$ ) in the T direction. Such a polynomial is obviously a sum of homogeneous polynomials enjoying the same property, therefore

f(x,y)=\phi_{0}(x,y)+\phi_{1}(x,y)+\ldots..

Or $\phi_{k}(x,y)=c_{0}x^{k}+c_{1}x^{k-1}y+\ldots+c_{k}y^{k}$ is a homogeneous polynomial of degree $k$ .

It is easily found that if $k<\min\left(m+n+2,m^{\prime}+n^{\prime}+2\right)$ the polynomial $\phi_{k}(x,y)$ is completely arbitrary. Otherwise there is

\frac{2k-m-n-m^{\prime}-n^{\prime}-4+|k-m-n-2|+\left|k-m^{\prime}-n^{\prime}-2\right|}{2}

conditions for determining the coefficients $c_{0},c_{1},\ldots,c_{k}$ . These conditions are

\begin{gathered}c_{n+1}=c_{n+2}=\ldots=c_{k-m-1}=0\\ c_{n^{\prime}+1}^{\prime}=c_{n^{\prime}+2}^{\prime}=\ldots=c_{k-m^{\prime}-1}^{\prime}=0\end{gathered}

Or $\phi_{k}\left(\alpha x^{\prime}+\beta y^{\prime},\gamma x^{\prime}+\delta y^{\prime}\right)=c_{0}^{\prime}x^{\prime k}+c_{1}^{\prime}x^{\prime k-1}y^{\prime}+\ldots+c_{k}^{\prime}y^{\prime k}$ .
We can also write the hypotheses made on $f(x,y)$ in the form

\frac{\partial^{m+n+2}f(x,y)}{\partial x^{m+1}\partial y^{n+1}}\equiv 0,\left(\alpha\frac{\partial}{\partial i}+\gamma\frac{\partial}{\partial y}\right)^{\left(m^{\prime}+1\right)}\left(\beta\frac{\partial}{\partial x}+\delta\frac{\partial}{\partial y}\right)^{\left(n^{\prime}+1\right)}f(x,y)\equiv 0

(66)

From these relationships we easily deduce that

\frac{\partial^{m+n+m^{\prime}+n^{\prime}+3}f(x,y)}{\partial x^{i}\partial y^{m+n+m^{\prime}+n^{\prime}+3-i}}\equiv 0,\quad i=0,1,\ldots,m+n+m^{\prime}+n^{\prime}+3

so,
Any polynomial that is a pseudo-polynomial of order ( $m,n$ ) in the direction $\mathrm{T}_{0}$ and a pseudo-polynomial of order ( $m^{\prime},n^{\prime}$ ) in the direction T, is at most of degree $m+n+m^{\prime}+n^{\prime}+2$ .

Let's suppose $m^{\prime}=m,n^{\prime}=n$ . From (66) we easily deduce that for that $f(x,y)$ be a pseudo-polynomial of order ( $m,n$ ) in any
direction it is necessary and sufficient that

\frac{\partial^{m+n+2}f(x,y)}{\partial x^{i}\partial y^{m+n+2-i}}\equiv 0,\quad i=0,1,\ldots,m+n+2

therefore,
The necessary and sufficient condition for a bounded function, or for a measurable function, to be a pseudo-polynomial of order ( $m,n$ ) in any direction is that this function is any polynomial of degree $m+n+1$ .
35. - There may exist polynomials of degree $>m+n+1$ which are pseudo-polynomials of order ( $m,n$ ) in an infinity of different directions. We will say that the direction 'T is rectangular if the transformation (63) is orthogonal, therefore if the axes Ox'y' are rectangular (the primitive axes Oxy are by hypothesis).

So that $f(x,y)$ be a pseudo-polynomial of order ( $m,n$ ) in any rectangular direction, it is necessary and sufficient that we have

		$\displaystyle\sum_{r=0}^{n+1}(-1)^{r}\binom{n+1}{r}\binom{m+1}{i-r}\frac{\partial^{m+n+2}f(x,y)}{\partial x^{m+1+i-2r}\partial y^{n+1-i+2r}}\equiv 0$
		$\displaystyle\quad i\quad 1,\ldots,m+n+2,\quad\binom{p}{q}=0\quad\text{ si }\quad p<q$

We immediately deduce that

\frac{\partial^{m+n+3}f(x,y)}{\partial x^{i}\partial y^{m+n+3-i}}\equiv 0,\quad i=0,1,\ldots,m+n+3

so that $f(x,y)$ is at the highest degree $m+n+2$ . This polynomial is therefore of the form $\mathrm{P}(x,y)+\phi_{m+n+2}(x,y)$ , Or $\mathrm{P}(x,y)$ is an arbitrary polynomial of degree $m+n+1$ . To determine the polynomial $\phi_{m+n+2}$ , it is advantageous to write it in the form $\phi_{m+n+2}(x,y)==\sum_{j=0}^{m+n+2}c_{j}z^{m+n+2-j}\bar{z}j$ , Or $z=x+iy$ with $i=\overline{-1}$ And $\alpha$ denotes the imaginary number conjugate of $\alpha$ . We heard of course, $c_{j}=\bar{c}_{m+n+2+j}$ . This polynomial must verify the equation

\left(\frac{\partial}{\partial z}+t\frac{\partial}{\partial z}\right)^{(m+1)}\left(\frac{\partial}{\partial z}-t\frac{\partial}{\partial z}\right)^{(n+1)}\phi_{m+n+2}=0

regardless of $t$ . We find that

c_{j}\sum_{r=0}^{n+1}(-1)^{r}\binom{n+1}{r}\binom{m+1}{j-r}\quad 0,\quad j=0,1,\ldots,m+n+2

(67)

The coefficient $c_{j}$ is zero or arbitrary depending on whether the second factor of (67) is $\neq 0$ Or $=0$ . It may happen that $\phi_{m+n+2}$ disappears completely. Otherwise, we see that it is always divisible by $x^{2}+y^{2}$ . The following table shows us the results for some values of $m$ And $n$

$\frac{n}{0}$		$\phi_{m+n+2}$
	${}_{\text{impair }}$
0	peer	$\mathrm{C}\left(x^{2}+y^{2}\right)^{\frac{m+2}{2}}$
1	$m$ +. 3 different from a perfect square	0
1	$m+3=p^{2}$	$\left(x^{2}+y^{2}\right)^{\frac{p^{2}-p}{2}}\left\lfloor cz^{p}+\bar{c}\bar{z}^{p}\right]$
2	odd, $3m+10$ different from a perfect square	0
2	peer, $3m+10$ different from a perfect square	$\mathrm{C}\left(x^{2}+y^{2}\right)^{\frac{m+4}{2}}$
2	odd, $3m+10=p^{2}$	$\left(x^{2}+y^{2}\right)^{\frac{(p-1)}{2}(\overline{p-2})}\left(cz^{P}+\bar{c}\bar{z}^{P}\right)$
2	peer, $3m+10=p^{2}$	$\mathrm{C}\left(x^{2}+y^{2}\right)^{\frac{m+4}{2}}+\left(x^{2}+y^{2}\right)^{\frac{(p-1)(p-2)}{2}\left(cz^{p}+\bar{c}\bar{z}^{p}\right)}$
	$m=n$	$\sum_{j=0}^{n}\mathrm{C}_{j}x^{n-j}y^{n-j}\left[x^{2j+2}+(-1)^{j}y^{2j+2}\right]$

Here $\mathrm{C},\mathrm{C}_{1},\ldots$ are real constants and $c$ an arbitrary complex constant.
36. - Let us make some remarks on the previous results. The pseudo-polynomial $f(x,y)$ of No. 33 was actually subject to the sole condition of verifying the equations $\Delta_{\beta h,-ah}^{m^{\prime},n^{\prime}}f_{1}\left(x^{\prime},y^{\prime}\right)=0$ , $\Delta_{\delta h,-\gamma h}^{m^{\prime},n^{\prime}}f_{1}\left(x^{\prime},y^{\prime}\right)=0$ . If, more generally, we assume that $f_{1}\left(x^{\prime},y\right)$ verifies a certain equation of the first kind, $\Delta_{h}^{m^{\prime},n^{\prime}}f_{1}\left(x^{\prime},y^{\prime}\right)=0$ , we obtain, at least in part, the same results. The only difference is that the equations verified by the
coefficients $A,B$ are of a slightly more general form than (64), (65), but are, in any case, of the form (28) (more exactly of the form (29). We can therefore state the following property:

The most general linearly measurable function, including a difference of the second kind in one direction $\mathrm{T}_{1}$ and a difference of the first kind in one direction ${}^{\mathrm{T}}{}_{2}$ , completely distinct from $\mathrm{T}_{1}$ , are identically zero, is a polynomial.

The form of this polynomial is, moreover, obviously the one found above.

We can assume the function is (superficially) measurable and the result still remains. Indeed, it is easily demonstrated that the function is linearly measurable. Let $\mathrm{E}_{x}$ the whole (measurement) $b_{1}-a_{1}$ ) of the $x$ for which $f(x,y)$ eats a measurable function of $y$ . Either $x_{0}$ a value of $x$ and let us take into account the fact that $f_{i}\left(x^{\prime},y^{\prime}\right)$ check the equation $\Delta_{h}^{m^{\prime},n^{\prime}}f_{i}\left(x^{\prime},y^{\prime}\right)=0$ Let's vary $x^{\prime}$ And $y^{\prime}$ so that we constantly have $x_{0}=\alpha x^{\prime}+\beta y^{\prime}$ . For any pair of such values of $x^{\prime},y^{\prime}$ , the numbers $x_{0}+(\alpha i+\beta j)h$ , $i+j>0$ belong to $\mathrm{E}_{x}$ , for almost all values of h. In particular, this is true for values as small as one wants of $h$ , which is enough to conclude that $f(x,y)$ is measurable by $y$ for everything $x$ . We demonstrate in the same way that $f(x,y)$ is a measurable function of $x$ for any value of $y$ . Note that, in reality, the function $f(x,y)$ verifies a system of two equations of the form (51). The second of these equations is precisely of the form indicated at the end of No. 3I, which ensures linear measurability.

We could further generalize the previous result, assuming that in the first direction only a difference of the first kind is identically zero. Of course, we assume that the function still satisfies certain conditions, under which we can affirm that it is a pseudo-polynomial.

Let us designate by $f(\mathrm{M})$ the value of the function at the point $\mathrm{M}(x,y)$ : MN Ciorinnescu [3] demonstrated that if the function verifies the functional equation

f(\mathrm{M})+f\left(\mathrm{M}_{1}\right)=f\left(\mathrm{M}_{2}\right)+f\left(\mathrm{M}_{3}\right)

(68)

for any quadruple of points $\mathrm{M},\mathrm{M}_{1},\mathrm{M}_{2},\mathrm{M}_{3}$ forming a rectangle ( $\mathrm{M},\mathrm{M}_{1}$ opposite summits', it is reduced to

\mathrm{C}\left(x^{2}+y\right)+\mathrm{C}_{1}x+\mathrm{C}_{2}y+\mathrm{C}_{3}.

MN Cioranescu assumes the existence of derivatives of two first
orders of $f(x,y)$ . We see that the result of MN Crorxnescu remains under much more general hypotheses. For example, under the following hypotheses:
$1^{0}.f(x,y)$ is a measurable function.
$2^{0}$ . equation (68) remains if $\mathrm{M},\mathrm{M}_{1},\mathrm{M}_{2},\mathrm{M}_{3}$ form a rectangle with sides parallel to the Oxy axes.
$3^{0}$ . equation (68) also remains if $\mathrm{M},\mathrm{M}_{1},\mathrm{M}_{2},\mathrm{M}_{3}$ form a square with sides parallel to the axes of a rectangular direction $\mathrm{T}^{\prime}$ , completely distinct from ' $\Gamma_{0}$ .

Indeed, $f(x,y)$ must be of the form $\mathrm{C}x^{2}+\mathrm{C}^{\prime}y^{2}+\mathrm{C}_{3}x+\mathrm{C}_{2}v+\mathrm{C}_{3}$ and condition 30 still gives $\mathrm{C}=\mathrm{C}^{\prime}$ .

BIBLIOGRAPHY

1.

Anohelutza Th. «On a functional equation characterizing polynomials» Mathematica, vol. VI (1932), pp. 1-7.
2 Brouwer LEJ «Orer differentiequotienten en differentialquotienten» Verhandelingen der K. Akad. van Wetenschappen Amsterdam, l. 17 (1908), pp. 38-45.
2.

Croranescu N. «A characteristic property of the rotation paraboloid» Bulletin of Math. and Ph. of the Ec. Polyt. Bucharest, VI amée, Fasc. 16-18 (1936) p. 134-135.

4 Darboux G. “On the composition of forces in statics” Bulletin des Sciences Matlı., t. 9 (1875) p. 281-288.
5. Densoy A. “On the integration of higher order differential coefficients” Fundamenta Mathematicae, L. XXV (1935), p. 273-326.
6. Fubni G. “Sugli integrali multipli” Rendiconti Academia dei Lincei, ser. 5 v. 16 (I sem.) (1907), p. 608-614,

8. Marchaud A. «On the derivatives and on the differences of functions of real variables» Jouruil de Math. L. VI (1927), pp. 332-425.
9 Pream E. «Legons on some functional equations» Paris, Gatulher-Villars, 1928.
10. Popovicu Tib. «On some properties of functions of one or two real variables» Thesis, Paris, 1933 or Mathematica, l. VIII (1934), pp. 1-86.
11. - «On certain functional equations defining polynomials» Mathematica, t. X. (1934) pp. 197-211.
12. Sierpingsi W. «On measurable converses» Fundamenta Mathematicae, t. I (1920', p. 125-129.

On bounded solutions and measurable solutions of certain functional equations

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Paper (preprint) in HTML form

ON BOUNDED SOLUTIONS AND MEASURABLE SOLUTIONS OF CERTAIN FUNCTIONAL EQUATIONS

SUMMARY

Introduction

CHAPTER I

Notations and some preliminary properties

We call the expression

$\coprod_{s=r}^{m}\prod_{i=1}^{n_{j}}\left(1-x_{j_{s}}^{\alpha_{j}s_{i}}\right)$ and we see that

CHAPTER II

On a class of functional equations with one variable

CHAPTER III

On pseudo-polynomials of two or more variables

CHAPTER IV

On some functional equations with several independent variables

CHAPTER V

On some functional properties characterizing polynomials of two variables

BIBLIOGRAPHY

Related Posts

On bounded solutions and measurable solutions of certain functional equations

Abstract

Authors

Keywords

Paper coordinates

PDF

About this paper

Journal

Publisher Name

DOI

Print ISSN

Online ISSN

References

Paper (preprint) in HTML form

ON BOUNDED SOLUTIONS AND MEASURABLE SOLUTIONS OF CERTAIN FUNCTIONAL EQUATIONS

SUMMARY

Introduction

CHAPTER I

Notations and some preliminary properties

We call the expression

∐s=rm∏i=1nI(1−xIsαI​si)\coprod_{s=r}^{m}\prod_{i=1}^{n_{j}}\left(1-x_{j_{s}}^{\alpha_{j}s_{i}}\right)and we see that

CHAPTER II

On a class of functional equations with one variable

CHAPTER III

On pseudo-polynomials of two or more variables

CHAPTER IV

On some functional equations with several independent variables

CHAPTER V

On some functional properties characterizing polynomials of two variables

BIBLIOGRAPHY

Related Posts

$\coprod_{s=r}^{m}\prod_{i=1}^{n_{j}}\left(1-x_{j_{s}}^{\alpha_{j}s_{i}}\right)$ and we see that