1. 1.
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     Let
     (1)

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a finite or infinite sequence of real, uniform functions defined in the finite and closed interval ($a,b$).

We will call it a linear combination of order$n$an expression of form

where the$c_{l}$are constants (which can also be zero).
We will say that the functions (1) form a basis when a linear combination of order$n$is determined completely by its values ​​on$n+1$distinct points and this whatever$n$and the$n+1$points considered.

Let's ask

For functions (1) to form a basis it is necessary and sufficient that we have$V\binom{f_{0},f_{1},\ldots,f_{n}}{x_{1},x_{2},\ldots,x_{n+1}}\neq 0$for all groups of$n+1$
distinct points$x_{1},x_{2},\ldots,x_{n+1}$of the interval ($a,b$) and for$n=0,1,2,\ldots$

In particular, the functions

form a base. In this case the determinant (2) becomes

and is none other than the Van Der Monde determinant of quantities$x_{1},x_{2},\ldots,x_{n+1}$. The base (3) is also the simplest that we can consider ( ¹ ).

We will also say that the functions (1) form a Tahebychef system, or a system (T) if, in particular, we have ( ² ).

When functions (1) are continuous, determinant (2) cannot change sign without canceling. We can therefore say that if functions (1) are continuous and if they form a basis, they also form a system (T), possibly changing the sign of some of these functions$\left({}^{3}\right)$.
(1) From a sequence (1) we can deduce an infinity of others
(a)

Or$\phi_{n}(x)$is a linear combination of order$n$, this order being effective, that is to say that the coefficient$c_{n}\neq 0$. If the functions (1) form a basis, the same is true for the functions ($\alpha$.) and vice versa. We can regard all these sequences as equivalent.
(2) The essential thing in this definition is that the determinant (4) does not vanish and does not change sign for a given n. If a sequence (1) verifies this property we can deduce a system (T) from it according to the definition in the text, possibly changing the sign of some of these functions. Such changes of signs are of no importance to us.
(3) This property may not be true if the functions (1) are not continuous, e.g., the functions

form a base but not a system (T).

For example, the functions (3) form a system (T), which is also the most important.

In the following we only consider systems (T).
2. - Consider a function$f(x)$defined on a set E whose points are part of the interval ($a,b$). Let us designate by$\mathrm{P}\left(\begin{array}[]{l|l}f_{0},f_{1},\ldots,f_{n}&f;x)\text{ la combinaison linéaire d'ordre }n\text{ qui }\\ x_{1},x_{2},\ldots,x_{n+1}&\end{array}\right.$takes the values$f\left(x_{l}\right)$to the points$x$

We will say that$f(x)$is non-concave, convex, polynomial, non-convex or concave of order n with respect to the functions$f_{0},f_{1},\ldots,f_{n}$depending on whether one of the inequalities

is verified on the whole set E .
We will also say that such a function is of order$n$in relation to functions$f_{0},f_{1},\ldots,f_{n}$. It is therefore a function for which the first member of relation (5) does not change sign on the set E. In the following we delete the words "with respect to the functions . . ." whenever this does not cause confusion.

The previous definition is a geometric definition, but we can transform it into another analytical one which will better highlight the symmetry of inequality (5) with respect to the points$x_{i}$.

We can write

Condition (5), taking into account (4), becomes

Note that condition (6) can still be written

and in this form we see that analytically the character of convexity is written independently of the order of the points$x_{i}$,

The non-concave function of order$n$can be considered as a type of function of order n. Convexity and polynomiality are special cases of non-concavity and non-convexity is obtained by a change of sign.

In short, the convexity (or concavity) of order$n$of the function$f(x)$expresses this property that functions$f_{0},f_{1},\ldots,f_{n}$And$f(x)$(or the function -$f(x)$) form a system (T). For a system (T) any function in the sequence is convex with respect to the others preceding it.

In the case of the sequence (3) we have the usual functions of order$n$In this case the first member of relation (7) is also written$\left[x_{1},x_{2},\ldots,x_{n+2};f\right]$and is what is called the divided difference of order$n+1$of the function on the points$x_{i}\left({}^{4}\right)$.
3. - A polynomial function is reduced to the values ​​on E of a linear combination of order$n$.

If$f(x)$is an order function$n$and if

it is polynomial of order$n$on the part of E included in the smallest interval containing the points$x_{l}$. Indeed, let us suppose that$x_{1}<x_{2}<\ldots<x_{n+2}$. The property results from the fact that the function$f(x)$remains included, in the interval ($x_{1},x_{n+4}$), between the two linear combinations of order$n$

which, as a result of equality (8), coincide,
( ⁴ ) See: Tiberiu Popoviciu „On some properties of functions of one and two real variables". Mathematica t. Vlll, p. 1-85. In the following we constantly refer to this work.
4. - In a general way if$x_{1}<x_{2}<\ldots<x_{n+2}$are points of E and$f(x)$is an order function$n$, it remains included, in the interval ($x_{1},x_{n+2}$), between the two linear combinations ($y$). We easily deduce the following property:

For any order function$n$is bounded on any subset completely interior to E it is necessary and sufficient that the functions$f_{0},f_{1},\ldots,f_{n}$are bounded on any subset completely interior to E.

A subset of$\mathbf{E}$is completely internal to$\mathbf{E}$if its ends are different from those of E. We call the ends of the set E its lower bound and its upper bound.

The condition is obviously necessary since the functions$f_{0},f_{1},\ldots,f_{n}$are of order$n$.

If the set E contains its endpoints, the property is true for the entire set E.

In particular, we deduce the following property:
For all the functions of a system (T) to be bounded it is necessary and sufficient that the first function$t_{0}(x)$is bounded.
5. - Suppose that$f_{0},f_{1},\ldots,f_{n}(n\geq 1)$are continuous at a point$x$which belongs at the same time to E and to its derivative$\mathrm{E}^{\prime}$.

Let's take two groups of$n$points

distinct from each other and from$x$, belonging to$E$in such a way that if there are any$i$And$i^{\prime}$points to the left of$x$in both groups (10) and (11) respectively, the number$i+i^{\prime}$is odd. We can always obtain this arrangement if we assume that$x$does not coincide with an end of E.

To fix the ideas we assume that

Be it now$f(x)$an order function$n$And$x^{\prime}$a point of E neighboring$x$. If we apply inequality (6) to sequences of points

$x_{1}^{\prime}<x<x^{\prime}<\left(\right.$Or$\left.x_{1}^{\prime}<x^{\prime}<x<\right)\quad x_{2}^{\prime}<x_{3}^{\prime}<\ldots<x_{n}^{\prime}$
we easily find that

where A, B are two quantities tending to zero when$x\rightarrow x^{\prime}$in an arbitrary manner.

We deduce that$f(x)$is continuous at the point$x$.
We therefore have the following property:
For any function of order n to be continuous on any subset completely interior to E it is necessary and sufficient that the functions$f_{0},f_{1},\ldots,f_{n}$are continuous on any subset completely interior to E.

We cannot, of course, say anything about the ends of the set E.

We have, in particular, the property:
For all functions of a system (T) to be continuous in the open interval ($\sigma,b$) it is necessary and sufficient that the first two functions$f_{0}(x),f_{1}(x)$are continuous in this interval.
6. - Let the two sequences of points (10), (11). Suppose that an interval ($c,d$) does not contain any points (10)," (11) and that there are$i,i^{\prime}$points to the left of$c$in the two groups (10), (11) respectively, the sum$i+i^{\prime}$being odd. To clarify the ideas, we will still assume that

Now consider a function$f(x)$of order$n$and two points of$\mathrm{E}x<x^{\prime}$located in the interval ($c,d$).

Inequalities

divided by$x^{\prime}-x$give us

Now suppose that the functions$f_{0},f_{1},\ldots,f_{n}$are with first bounded divided difference on any subset completely interior to E. They are therefore, as well as the function$f(x)$, bounded and continuous on any subset completely interior to E. On the other hand,$x$remaining in the interval ($c,d$), the quantities

have a positive lower bound.
Finally, we see that we have the following property:
For any function of order$n(n\geq 1)$is with first divided difference bounded on any subset completely interior to . E, it is necessary and sufficient that it is so for the functions$f_{0},f_{4},\ldots,f_{n}$.

The condition for a function to have bounded first divided difference is none other than the ordinary Lipschitz condition.

It should be noted that our demonstration shows that the subset considered cannot be completely arbitrary. It must exist at least$n+1$points of E not located in the interval formed by the ends of the subset in question and not all on the same side of this interval.

In particular:
For all functions of a system (T) to have a first divided difference bounded in any interval completely interior to ($a,b$) it is necessary and sufficient that it be so for the first two functions$f_{0}(x),f_{1}(x)$.

Let us first specify the assumptions that we are going to make about the system (T).

Let us extend the meaning of determinant (2) to the case where the points$x_{i}$are not all distinct. Generally speaking, if in

points$x_{i_{1}},x_{i_{2}},\ldots,x_{i_{j}}$are confused at one point$x$all the others$x_{i}$being distinct from$x$, the symbol means the determinant
(2) where the rows of rank$i_{1},i_{2},\ldots,i_{j}$are replaced by the lines: