On equivalence of moduli of smoothness
type: Item
instance of: scholarly article
title: On equivalence of moduli of smoothness
zbMATH Open document ID: 0953.41018
published in: Journal of Approximation Theory
publication date: 2000-08-27T00:00:00Z
review text: It is known that if \(f \in W^k_p\), then \[ \omega_m(f,t)_p \leq t \omega_{m-1}(f',t)_p \leq \cdots. \] The inverse statement with constants independent of \(f\) is not true in general. \textit{Y. Hu} and \textit{X. Yu} [SIAM J. Numer. Anal. 32, No. 5, 1428-1435 (1995; Zbl 0841.41007)] proved that the inverse holds true for splines \(S\) with equally spaced knots; thus , \[ \omega_m(S,t)_p \sim t \omega_{m-1}(S',t)_p \sim t^2 \omega_{m-2}(f'',t)_p \cdots \] In this paper the authors extend Hu and Yu's results to splines with any given knot sequence, and further to principal shift--invariant spaces and wavelets under certain conditions. Some applications are also given.
Mathematics Subject Classification ID: 41A50
Mathematics Subject Classification ID: 41A15
Mathematics Subject Classification ID: 42C40
zbMATH DE Number: 1278845
zbMATH Keywords: modulus of smoothness
zbMATH Keywords: splines
zbMATH Keywords: shift-invariant spaces
zbMATH Keywords: wavelets
zbMATH Keywords: degree of approximation
zbMATH Keywords: convex approximation
author: Yingkang Hu
reviewed by: Richard A. Zalik
MaRDI profile type: Publication
OpenAlex ID: W2054293364
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DOI: 10.1006/JATH.1997.3264
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description: scientific article; zbMATH DE number 1278845