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

O.V. Davydov (Dnipropetrovsk State University), https://orcid.org/0000-0001-8813-9485

#### 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.

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#### References

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DOI: https://doi.org/10.15421/248704

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