Zentralblatt MATH
Publications of (and about) Paul Erdös
Zbl.No: 692.41004
Autor: Erdös, Paul; Kroò, A.; Szabados, J.
Title: On convergent interpolatory polynomials. (In English)
Source: J. Approximation Theory 58, No.2, 232-241 (1989).
Review: Let Xn: -1 \leq xnn < xn-1,n < ... < x1n \leq 1 (n = 1,2,...) be a system of nodes of interpolation. Let xkn = \cos tkn, 0 \leq t1n < t2n < ... < tnn \leq \pi, and for an arbitrary interval I\subseteq [0,\pi], denote Nn(I) = sumt_{kn in I}1. Let \Pim 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 ||·|| 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 pn(x) in \Pi[n(1+\epsilon)] such that pn(xkn) = f(xkn) (k = 1,...,n; n = 1,2,...) and ||f(x)-pn(x)|| = O(E[n(1+\epsilon)](f)) hold if and only if limsupn ––> oo Nn(In)/n|In| \leq 1/\pi whenever limn ––> oon |In| = oo (|In| = length of In) and liminfn ––> oo max1 \leq i \leq n-1 n(ti+1,n-ti,n) > 0. Here the sign O refers to n ––> oo and indicates a constant depending only on \epsilon; En(f) is the best uniform approximation of f(x) by polynomials of degree at most n.
Reviewer: Y.Sitaraman
Classif.: * 41A05 Interpolation
41A10 Approximation by polynomials
Keywords: nodes of interpolation; algebraic polynomials
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