Some conditions of convergence of interpolative Lagrange processes on $$$A_R$$$ and $$$\mathbb{C}^{\infty}$$$ classes

Abstract

Let $$$X = \{ -1 \leqslant x_{0n} < \ldots < x_{nn} \leqslant 1;\; n=0,1,2,\ldots \}$$$ be a matrix of nodes; $$$L_n(X, f)$$$ be the interpolative Lagrange process; $$$\lambda_n(X)$$$ be the Lebesgue constant; $$$A_R$$$ be the class of functions that are regular in ellipse with the sum of semiaxes being equal to $$$R > 1$$$. Let
$$R_0(X) = \inf \bigl\{ R > 1\colon \forall f \in A_R \lim\limits_{n \rightarrow \infty} \| f - L_n(X, f) \| = 0 \bigr\}$$
Theorem 1. Let the nodes of the matrix $$$X$$$ satisfy the condition $$$| \theta_{in} - \theta_{i-1,n}| \geqslant \frac{\varepsilon \pi}{n}$$$, $$$i = \overline{1, n}$$$, where $$$\theta_{in} = \arccos x_{in}$$$, $$$n = 1, 2, \ldots$$$, $$$0 < \varepsilon \leqslant 1$$$. Then the following inequality holds:
$$\bigl( \lim\limits_{n \rightarrow \infty} \sqrt[n]{\lambda_n(X)} \bigr)^{\varepsilon} \leqslant R_0(X) \leqslant \lim\limits_{n \rightarrow \infty} \sqrt[n]{\lambda_n(X)}$$
Analogous results take place for the classes $$$A_R$$$ of all regular and infinitely differentiable on $$$\mathbb{C}^{\infty}$$$ functions.