Minimal number of significant directional moduli of smoothness
type: Item
instance of: scholarly article
title: Minimal number of significant directional moduli of smoothness
zbMATH Open document ID: 0791.41031
DOI: 10.1007/BF01904036
author: Z. Ditzian
author: Kamen Ivanov
published in: Analysis Mathematica
publication date: 1994-07-07T00:00:00Z
review text: Let \(\Delta^ k_{he} f(x)\) be the \(k\)th finite difference of the function \(f(x)\), \(x\in D\subset\mathbb{R}^ d\), with respect to the domain \(D\), that is \[ \Delta^ k_{he} f(x)=\begin{cases} 0\quad\text{if Conv-hull } (x,x+khe)\not\subset D,\\ \sum^ k_{m=0} (-1)^{k-m}{k\choose m} f(x+ mhe)\quad\text{otherwise}.\end{cases} \] The directional \(k\)th modulus of smoothness for \(L_ p(D)\) is defined by \[ \omega^ k(f,e,t)_{L_ p(D)}=\sup_{| h|\leq t}\|\Delta^ k_{he} f(x)\|_{L_ p(D)}. \] The vector \(e\) is called the direction of \(\omega^ k(f,e,t)\). Then the modulus of smoothness is defined by \[ \omega^ k(f,t)_{L_ p(D)}= \sup_{| e|=1} \omega^ k(f,e,t)_{L_ p(D)}. \] In this paper the authors show that there exist \(n\) directions \(e_ 1,e_ 2,\dots,e_ n\) such that \[ \omega^ k(f,t)_{L_ p(D)}\leq C(k,p,D,n) \sup_{1\leq m\leq n} \omega^ k(f,e_ m,t)_{L_ p(D)}. \] The obtained results cannot be improved.
Mathematics Subject Classification ID: 41A65
Mathematics Subject Classification ID: 41A17
zbMATH DE Number: 500208
zbMATH Keywords: modulus of smoothness
reviewed by: Borislav Crstici
MaRDI profile type: Publication
cites work: Moduli of Continuity in R n and D ⊂R n
cites work: A priori estimates for differential operators in \(L_ \infty\) norm
cites work: A non-inequality for differential operators in the \(L_ 1\) norm
cites work: Q5607484
full work available at URL: https://doi.org/10.1007/bf01904036
OpenAlex ID: W35352343
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description: scientific article; zbMATH DE number 500208