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

Let $X = \{ -1 \leqslant x_{0n} 1$. 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.