On convergent interpolatory polynomials
Let \(X_n: -1 \leq x_{nn}<x_{n-1,n}<...<x_{1n}\leq 1\) \((n=1,2,...)\) be a system of nodes of interpolation. Let \(x_{kn}=\cos t_{kn}\), \(0\leq t_{1n}<t_{2n}<...<t_{nn}\leq \pi\), and for an arbitrary interval \(I\subseteq [0,\pi]\), denote \(N_n(I)=\sum_{t_{kn}\in I}1\). Let \(\Pi_m\) be the set of algebraic polynomials of degree at most \(m\), \(C[-1,1]\) be the space of continuous functions on the interval \([-1,1]\), and \(||\cdot||\) be the maximum norm over \([-1,1]\). The following theorem is proved: Theorem. For every \(f(x)\in C[-1,1]\) and \(\epsilon >0\) there exists a sequence of polynomials \(p_n(x)\in \Pi_{[n(1+\epsilon)]}\) such that \(p_n(x_{kn})=f(x_{kn})\) \((k=1,...,n\); \(n=1,2,...)\) and \(||f(x)-p_n(x)|| = O(E_{[n(1+\epsilon)]}(f))\) hold if and only if \(\varlimsup_{n\to \infty} N_n(I_n)/n|I_n| \leq 1/\pi\) whenever \(\lim_{n\to \infty}n |I_n| = \infty\) \((|I_n| =\) length of \(I_n)\) and \(\varliminf_{n\to \infty} \min_{1\leq i\leq n-1} n(t_{i+1,n}-t_{i,n}) > 0.\) Here the sign O refers to \(n\to \infty\) and indicates a constant depending only on \(\epsilon\); \(E_n(f)\) is the best uniform approximation of \(f(x)\) by polynomials of degree at most \(n\).
scientific article; zbMATH DE number 4132804