Jackson's theorem for compact connected Lie groups
type: Item
instance of: scholarly article
title: Jackson's theorem for compact connected Lie groups
zbMATH Open document ID: 0704.43007
DOI: 10.1016/0021-9045(88)90102-5
author: Donald I. Cartwright
author: Krzysztof Kucharski
published in: Journal of Approximation Theory
publication date: 1988-01-01T00:00:00Z
review text: The following theorem is obtained. Th: Let G be a CCLG (compact connected Lie group), \(E=C(G)\), \(L^ p(G)\), \(1\leq p<\infty\), \(r\in \{1,2,..\}\). Then for each \(n\in Z^+\) there exists a central trigonometric polynomial \(K_ n\) of degree \(\leq n\), such that \(\| f-K_ n*f\|_ E\leq C_ r\omega_ r(n^{-1},f)\), where \(\omega_ r(t,f)\) is the r-th modulus of continuity of f in E. The case \(r=2\) was known before [see \textit{H. Johnen}, Linear Operators and Approximation, ISNM 20, 254-272 (1972; Zbl 0266.43005)]. As a result, combining this result with the sharp estimates of the Lebesgue constants of Fourier partial sums [see \textit{S. Giulini}, \textit{G. Travaglini}, J. Funct. Anal. 68, 106-116 (1986; Zbl 0664.22008)], it is shown that: Let G be a semisimple CCLG, \(E=C(G)\), \(L^ 1(G)\), and \(f\in E\) such that \(\omega_ r(t,f)=o(t^{(d-1)/2})\), \(t\to 0\), for some \(r>(d-1)/2\), where \(d=\dim (G)\). Then the spherical partial sum \(S_ nf\to f\) in E. This might be said to be a ``best possible'' criterion.
Mathematics Subject Classification ID: 43A80
Mathematics Subject Classification ID: 43A15
Mathematics Subject Classification ID: 41A50
zbMATH DE Number: 4155161
zbMATH Keywords: compact connected Lie group
zbMATH Keywords: central trigonometric polynomial
zbMATH Keywords: modulus of continuity
zbMATH Keywords: Lebesgue constants
zbMATH Keywords: Fourier partial sums
reviewed by: Ruilin Long
MaRDI profile type: Publication
full work available at URL: 0021-9045(88)90102-5
OpenAlex ID: W2057824183
Wikidata QID: Q115365289
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description: scientific article; zbMATH DE number 4155161