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 {xk-ck}1oo. (In English)
Source: J. Approximation Theory 43, 75-80 (1985).
Review: Several easily verified conditions on a sequence (ck)1oo of real numbers are given which imply that the sequence of functions (xk-ck)1oo 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 (xk-ck)1oo in C[0,1]. For instance, if the sequence (ck)1oo is such that for all k \geq M, \epsilon(-1)k(ck-c) \geq 0, where c in R and \epsilon in {-1,1}, fixed, and if ck-c\not\equiv 0, then (xk-ck)1oo is total in C[0,1]; if, in addition, ck\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 xk-ck is given. Another condition is: |cnk-c|1/nk ––> 0 as k ––> oo, where the subsequence (nk)1oo satisfies the Müntz condition sumook = 1(nk)-1 = oo and ck\not\equiv c; in the case when |ck|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
© European Mathematical Society & FIZ Karlsruhe & Springer-Verlag