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**Zentralblatt MATH**

**Publications of (and about) Paul Erdös**

**Zbl.No: ** 576.41022

**Autor: ** Anderson, J.M.; Erdös, Paul; Pinkus, Allan; Shisha, Oved

**Title: ** The closed linear span of **{**x^{k}-c_{k}**}**_{1}^{oo}. (In English)

**Source: ** J. Approximation Theory 43, 75-80 (1985).

**Review: ** Several easily verified conditions on a sequence (c_{k})_{1}^{oo} of real numbers are given which imply that the sequence of functions (x^{k}-c_{k})_{1}^{oo} is total in C[0,1]. This problem is equivalent to demanding that the function f(x) \equiv 1 belongs to the closed linear hull of (x^{k}-c_{k})_{1}^{oo} in C[0,1]. For instance, if the sequence (c_{k})_{1}^{oo} is such that for all k \geq M, \epsilon(-1)^{k}(c_{k}-c) \geq 0, where c in **R** and \epsilon in **{**-1,1**}**, fixed, and if c_{k}-c\not\equiv 0, then (x^{k}-c_{k})_{1}^{oo} is total in C[0,1]; if, in addition, c_{k}\ne c for infinitely many k, with the help of Chebyshev polynomials an effective approximation to f(x) \equiv 1 in C[0,1] by finite linear combinations of the x^{k}-c_{k} is given. Another condition is: |c_{nk}-c|^{1/nk} ––> 0 as k ––> oo, where the subsequence (n_{k})_{1}^{oo} satisfies the Müntz condition **sum**^{oo}_{k = 1}(n_{k})^{-1} = oo and c_{k}\not\equiv c; in the case when |c_{k}|^{1/k} ––> 0 as k ––> oo, again, a good approximation to f(x)\equiv 1 is explicitly constructed.

**Reviewer: ** F.Haslinger

**Classif.: ** * 41A65 Abstract approximation theory

**Keywords: ** Chebyshev polynomials; Müntz condition

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