On approximation by integral Müntz polynomials
type: Item
instance of: scholarly article
title: On approximation by integral Müntz polynomials
zbMATH Open document ID: 0578.41009
author: T. Kano
published in: Proceedings of the Japan Academy. Series A
publication date: 1983-01-01T00:00:00Z
review text: Let \(C_ 0[0,\alpha]\) be the set of all continuous functions f on the interval [0,\(\alpha\) ] such that f(m) is integer for any integer m in [0,\(\alpha\) ]. Let \(0=\lambda_ 0<\lambda_ 1<\lambda_ 2<..\). be a fixed sequence of real numbers and \(P_{\Lambda}\) the set of Müntz polynomials \(p(x)=\sum^{n}_{k=0}a_ k x^{\lambda_ k}\) with integer coefficients \(a_ k\). If the Müntz condition \(\sum 1/\lambda_ k=\infty\) is satisfied, then \(P_{\Lambda}\) is dense in \(C_ 0[0,1]\). On the other hand, it was shown by \textit{Le Baron O. Ferguson} [Fourier analysis and approximation theory, Vol. I, Budapest 1976, Colloq. Math. Soc. Janos Bolyai 19, 359-370 (1978; Zbl 0421.41005) that \(P_{\Lambda}\) is closed in \(C_ 0[0,\alpha]\) for \(\alpha >1\) if the sequence \(\Lambda\) is such that (1) \(\lambda_{k+1}-\lambda_ k\geq c>0\) and (2) lim sup N/\(\lambda\) \({}_ N=0\). Ferguson derived this result by means of a criterion which states that \(P_{\Lambda}\) is closed in \(C_ 0[0,\alpha]\) if (3) lim inf \(M_ n>0\), where \(M_ n\) are the norms of certain Chebyshev polynomials. The author of the paper under review cites this criterion but with an unjustified replacement of the lim inf by lim sup in (3). He then uses this modified criterion to replace in (2) the lim sup by a lim inf.
reviewed by: Rudolf Wegmann
Mathematics Subject Classification ID: 41A10
zbMATH DE Number: 3925454
zbMATH Keywords: Müntz polynomials
zbMATH Keywords: Müntz condition
MaRDI profile type: Publication
full work available at URL: pjaa.59.331
OpenAlex ID: W2072478082
cites work: Q3854091
cites work: Q3882893
cites work: Muntz-Szasz Theorem with Integral Coefficients. II
cites work: Q4103950
cites work: Q5515488
cites work: On sets of integers containing k elements in arithmetic progression
DOI: 10.3792/PJAA.59.331
Recommended article: Q3378727
Recommended article: Q4858543
Recommended article: Q3196729
Recommended article: Q4463398
Recommended article: Sur un procédé de sommation des séries trigonométriques.
Recommended article: Q5201720
Recommended article: Beiträge zur Theorie der fastperiodischen Funktionen. I: Funktionen einer Variablen.
Recommended article: Q3320654
Recommended article: Q3468091
Recommended article: Q5693273
description: scientific article; zbMATH DE number 3925454