By
TIBERIU POPOVICIU
(Cluj, Romanian RP)
(Received April 14, 1960.)
Dedicated to the memory of l. Fejér

We know the particular interest that L. Fejér has shown in problems of interpolation by polynomials. He himself has obtained results of great importance in this field. In the following I propose to make some very simple remarks on some of these problems.

1. 1.

   Consider the linear operator

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defined on the space of functions$f$, real and of a real variable$x$, defined on an interval$I$containing the distinct nodes
(2)

and where
(3)

are real functions of the real variable$x$, defined on the interval$I^{\prime}$. To fix the notations, we will always assume that
(4)

If, in particular, the (3) are polynomials we can take for$I^{\prime}$the real axis ($-\infty,\infty$).

By giving to$x$the value$x_{0}$we deduce from (1) the linear functional$\Phi\left[f\mid x_{0}\right]$.

We will say that operator (1) preserves the sign of the function$f$if we have$\Phi[f\mid x]\geqq 0$on$I^{\prime}$for any function$f$non-negative on$I$. For this to be so, it is necessary and sufficient that the linear functional$\Phi\left[f\mid x_{0}\right]$preserves the sign of the function$t$, so that it is non-negative for any function$t$non-negative on$I$and for everything$x_{0}\in I^{\prime}$.

It is always possible to construct a function$f$non-negative on$I$and taking on the nodes (2) any non-negative values. Such is, for example, any function represented by a suitable polygonal line joining the representative points corresponding to the nodes. It follows that the necessary and sufficient condition for the linear functional$\Phi\left[f\mid x_{0}\right]$preserves the sign of the function$f$is that the sequence (3) is non-negative for$x=x_{0}$.

16 Annals

We will say that operator (1) preserves the monotonicity of the function$f$if the function of$x,\Phi[f\mid x]$is non-decreasing on$I^{\prime}$for any function$f$non-decreasing on$I$.

Suppose we have

and that the functions (3) are differentiable on$I^{\prime}$. So, the derivative$\Phi^{\prime}[f\mid x]$of function (1) can be written

Note that we can always construct a function$f$non-decreasing on$I$and taking on the nodes (2) values ​​forming any non-decreasing sequence. Such is again, for example, any function represented by a suitable polygonal line joining the representative points corresponding to the nodes. It follows that the necessary and sufficient condition for (1), under the preceding hypotheses, to preserve the monotony of the function$f$is that the sequence of derivatives of functions (3) has all its non-positive partial sequences, so that we have

where for$i=n$equality is valid identically in$x$.
2. Suppose, in particular, that (1) is the Lagrange polynomial$L[f\mid x]$of the function$f$on the nodes (2). Then the functions (3) reduce to the fundamental interpolation polynomials

Or

We then have the following property:
I. If$n\geqq 3$and if$x_{0}$is different from the nodes (2), the linear functional$L\left[f\mid x_{0}\right]$does not preserve the sign of the function$f$.

Still assuming that condition (4) is verified, this property results only from the fact that the first 3 polynomials (8),$l_{1}(x),l_{2}(x),l_{3}(x)$,
cannot have the same sign (cannot all be$\geqq 0$or all of them$\leq 0)$For$x=x_{0}$. Indeed, we have

$x$being always different from the nodes.
We will also give another demonstration of property I. Let$x_{0}$different from the nodes and is a positive number small enough so that the closed interval$\left[x_{0}-a,x_{0}+a\right]$does not contain any nodes. Let us construct (which is always possible) a positive function$f$which takes the same values ​​as the polynomial of degree$2,\left(x-x_{0}\right)^{2}-a^{2}$on the nodes. We have$L\left[f\mid x_{0}\right]==L\left[\left(x-x_{0}\right)^{2}-a^{2}\mid x_{0}\right]=-a^{2}<0$and the property is demonstrated.

For$n=1$We have$L[f\mid x]==f\left(x_{1}\right)$and the linear functional$L\left[f\mid x_{0}\right]$preserves the sign of the function$f$for everything$x_{0}$.

For$n=2$We have

and the linear functional$L\left[f\mid x_{0}\right]$preserves the sign of the function$t$if and only if$x_{0}$belongs to the interval$\left[x_{1},x_{2}\right]$knots ($x_{1}<x_{2}$).
3. In the case (7) of the Lagrange polynomial, the equality (5) is indeed verified and the functions (3) (which are then polynomials) are everywhere differentiable.

We have the following property:
II. If$n\geqq 4$, the Lagrange polynomial$L[f\mid x]$does not maintain the monotony of the function$f$on any interval (non-zero)$I^{\prime}$.

We will give a demonstration analogous to the second demonstration of property I.

It is clear that it is sufficient to demonstrate the property for any interval$I^{\prime}$closed on the left. So then$x_{0}$the left end of$I^{\prime}$and, if$x_{0}<x_{n}$, either$x_{r}$. the first term of the sequence (2) located to the right of$x_{0}$(we have (4)). We can then always construct a function$f$non-decreasing taking the values, forming a non-decreasing sequence, of the polynomial of degree 2 or 3,

on the knots. But$L[P\mid x]=P(x)$and the polynomial$P$is not non-decreasing on$I^{\prime}$. On the contrary, this polynomial is decreasing

For$n=1$and for$n=2$property II is not true. In these cases we have$L^{\prime}[f\mid x]=0$And$L^{\prime}[f\mid x]=\frac{f\left(x_{2}\right)-f\left(x_{1}\right)}{x_{2}-x_{1}}$respectively and$L[f\mid x]$preserves the monotony of$f$over the entire interval$I^{\prime}$. If$n=3$, a simple calculation shows us that the conditions (6) for the conservation of monotony become

and we see that$L[f\mid x]$maintains the monotony of the function$f$on$I^{\prime}$if and only if this interval is a subinterval of

1. 4.

   Now suppose that (1) is the Fejér polynomial$F[f\mid x]$of the function$f$on the nodes (2). Then the functions (3) reduce to the fundamental Lagrange-Hermite interpolation polynomials of the first kind

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We know that then$F[f\mid x]$can well preserve the sign of the function$f$over the entire finite interval$I^{\prime}$.

Let's suppose that$I$is reduced to the interval$[-1,1]$. Then L. Fejér [1] demonstrated that if the polynomial$l(x)$checks the differential equation

ultraspherical polynomials and if$0\leq\lambda\leq 1$, the operator$F[f\mid x]$preserves the sign of the function$f$on the interval$[-1,1]$.

When$l(x)$verifies the differential equation (10) we will say that we are in the ultraspherical case of parameter$\lambda(\lambda>-1)$. Besides, if$\lambda>-1$the polynomial$l(x)$of degree$n$which verifies the differential equation (10) has all its real roots, distinct and included in the interval ($-1.1$). In particular, we are in the case of Legendre if$\lambda=1$and in the case of Tchebycheff if$\lambda=0$. Then the nodes are the roots of the Legendre polynomial resp. of the Tchebycheff polynomial of the first kind of degree$n$.

In particular therefore$F[f\mid x]$preserves the sign of the function$f$on the interval$[-1,1]$in the case of Legendre and in the case of Tchebycheff.
5. Still in the case of the operator$F[f\mid x]$of Fejér, suppose that the number$n=2m$of the nodes is even and that these nodes are distributed symmetrically with respect to the origin. We then have$l(0)\neq 0,l^{\prime}(0)=0,x_{2}=-x_{j},l_{2m+1-j}(0)=l_{j}(0)\neq 0,l^{\prime}\left(x_{2m+1-j}\right)=-l^{\prime}\left(x_{j}\right),l^{\prime\prime}\left(x_{2m+1-j}\right)=l^{\prime\prime}\left(x_{j}\right)$,$j=1,2,\ldots,m$And

We deduce that the$\varphi_{i}(0),i=1,2,\ldots,2m$are positive if and only if

We have the following property:
III. If the number$n=2m$of the nodes is even, if these nodes are symmetrically distributed with respect to the origin and if the inequalities (12) are verified, boperator$F[f\mid x]$de Fejér preserves the sign and also preserves the monotony of the hundred non-zero interval$[-a,a],(a>0)$, having its center in the origin.

The conservation of the sign results from the fact that we have$\varphi_{i}(0)>0,i=1$,$2,\ldots,2m$, as a result of (11) and the fact that the$\varphi_{i}(x)$are continuous functions.

To demonstrate the property relating to the conservation of monotony it is sufficient to demonstrate the inequalities (6) for$I^{\prime}=[-a,a]$. To be sure of the existence of such a number$a$it is sufficient, due to the continuity of functions$\varphi_{i}(x)$, to demonstrate that, in the case considered, we have

But, taking into account (11) and the reported symmetry of the knots, we deduce

from which inequalities (13) immediately follow.
Property III is therefore demonstrated.
6. In particular, if we are in the ultraspherical case the nodes are symmetrically distributed with respect to the origin.

Differential equation (10) shows us that in this case, if$\lambda>-1$,

so conditions (12) are verified.
It follows that we have the following property:
IV. If we are in the ultraspherical case of parameter$\lambda>-1$and if the number$n=2m$of nodes is even, the operator$F[f\mid x]$de Fejér preserves the sign and also preserves the monotony of the function$f$on a certain interval won null$[-a,a](a>0)$, having its center in the origin.

The property is true, in particular, in the case of Legendre and in the case of Tschebycheff.

The previous results are to be compared with the important property of conservation of convexities of all orders, which enjoy on the interval$[0,1]$SN Bernstein polynomials

which are also of the form (l). We formerly obtained these properties [2].