Zentralblatt MATH
Publications of (and about) Paul Erdös
Zbl.No: 767.41004
Autor: Erdös, Paul; Halász, G.
Title: On the arithmetic means of Lagrange interpolation. (In English)
Source: Approximation theory, Proc. Conf., Kecskemét/Hung. 1990, Colloq. Math. Soc. János Bolyai 58, 263-274 (1991).
Review: [For the entire collection see Zbl 746.00075.]
For f: [-1,+1] ––> R let pn(x) = Ln(f; x) be the Lagrange interpolation polynomial on the roots of the Chebyshev polynomial of degree n: pn(\cos \thetam,n) = f(\cos\thetam,n), m = 1,...,n, where \thetam,n = (2m-1)\pi/2n. The authors prove the following theorem: Given a sequence \lambda(N) ––> 0 however slowly, one can construct a continuous function f0(x) such that for almost all x in [-1,+1], (1/N)|sumn = 1N Ln(f0; x)| \geq \lambda(N) log log N, for infinitely many N.
This result corrects an oversight in the proof of a result of the first author and G. Grünwald [Studia Math. 7, 82-95 (1938; Zbl 018.11804)], that the analogue of Fejér's classical result about the arithmetic means of Fourier series is false for interpolation.
Reviewer: C.Mustata (Cluj-Napoca)
Classif.: * 41A05 Interpolation
Keywords: Lagrange interpolation polynomial; Chebyshev polynomial
Citations: Zbl 746.00075; Zbl 018.11804
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