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**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 **{**\xi_{n}**}** be a sequence on n points from [-1,+1] varying with n; let L_{n}(x) denote the sequence of Lagrange polynomials coinciding with a given R integrable function f(x) at the points \xi_{n}. The authors are interested in the mean convergence **lim**_{n ––> oo} **int**_{-1}^{+1} |f(x)-L_{n}(x)|^{p}\, dx = 0 (*) for p = 2 and p = 1. Let \xi_{n} be the zeros of the orthogonal polynomial p_{n}(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 \xi_{n} the zeros of p_{n}(x)+A_{n}p_{n-1}(x)+B_{n}p_{n-2}(x), where A_{n} arbitrary real, B_{n} \leq 0. If **int**_{-1}^{+1} w(x) dx and **int**_{-1}^{+1} w(x)^{-1}\, dx exist and \xi_{n} 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 **sum**_{k = 1}^{n} **int**_{-1}^{+1} l_{k}(x)^{2} \, dx is unbounded; here l_{k}(x) are the fundamental polynomials of the Lagrange interpolation corresponding to the set \xi_{n}.

**Reviewer: ** G.Szegö

**Classif.: ** * 41A05 Interpolation

42A15 Trigonometric interpolation

**Index Words: ** Approximation of functions, orthogonal series developments

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