Some relations between entropy and approximation numbers
type: Item
instance of: scholarly article
title: Some relations between entropy and approximation numbers
zbMATH Open document ID: 0961.47008
DOI: 10.1007/BF02882243
published in: Science in China. Series A
publication date: 2000-11-16T00:00:00Z
review text: The author relates the decay of the entropy numbers \(\varepsilon_k(T)\) of operators \(T\) in Hilbert spaces to the decay of their singular numbers \(s_n(T)\). He discusses various examples of exponential decay which is of different order for the singular and entropy numbers. He does not seem to be aware of the general formula \[ e_{k+ 1}(T)\sim \sup_{n\in\mathbb{N}} 2^{-k/n}(s_1(T)\cdots s_n(T))^{1/n} \] shown by \textit{Y. Gordon}, \textit{H. König} and \textit{C. Schütt} in [J. Approximation Theory 49, 219-239 (1987; Zbl 0647.47035)] from which his results can be deduced to a large extent.
Mathematics Subject Classification ID: 47B06
zbMATH DE Number: 1531226
zbMATH Keywords: entropy numbers
zbMATH Keywords: approximation numbers
zbMATH Keywords: Hilbert spaces
zbMATH Keywords: exponential decay
author: Zhiming Zheng
MaRDI profile type: Publication
cites work: Inequalities of Bernstein-Jackson-type and the degree of compactness of operators in Banach spaces
cites work: Inequalities between entropy and approximation numbers of compact maps
cites work: Relations between approximation numbers and entropy numbers
cites work: Q3995208
cites work: Infinite-dimensional dynamical systems in mechanics and physics
cites work: Q3764747
cites work: Inequalities between eigenvalues, entropy numbers, and related quantities of compact operators in Banach spaces
full work available at URL: https://doi.org/10.1007/bf02882243
OpenAlex ID: W2014553909
description: scientific article; zbMATH DE number 1531226