Zentralblatt MATH
Publications of (and about) Paul Erdös
Zbl.No: 016.10604
Autor: Erdös, Pál; Turán, Pál
Title: On interpolation. I. Quadrature- and mean-convergence in the Lagrange- interpolation. (In English)
Source: Ann. of Math., II. Ser. 38, 142-155 (1937).
Review: Let {\xin} be a sequence on n points from [-1,+1] varying with n; let Ln(x) denote the sequence of Lagrange polynomials coinciding with a given R integrable function f(x) at the points \xin. The authors are interested in the mean convergence limn ––> oo int-1+1 |f(x)-Ln(x)|p\, dx = 0 (*) for p = 2 and p = 1. Let \xin be the zeros of the orthogonal polynomial pn(x) of degree n corresponding to the weight function w(x) \geq \mu > 0. Then (*) holds with p = 2. The same is true if we choose for \xin the zeros of pn(x)+Anpn-1(x)+Bnpn-2(x), where An arbitrary real, Bn \leq 0. If int-1+1 w(x) dx and int-1+1 w(x)-1\, dx exist and \xin is defined by the zeros of the linear combination mentioned, (*) holds with p = 1. Finally the existence of a continous function f(x) is proved for which (*) with p = 2 does not hold provided that sumk = 1n int-1+1 lk(x)2 \, dx is unbounded; here lk(x) are the fundamental polynomials of the Lagrange interpolation corresponding to the set \xin.
Reviewer: G.Szegö
Classif.: * 41A05 Interpolation
42A15 Trigonometric interpolation
Index Words: Approximation of functions, orthogonal series developments
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