Improved converse theorems and fractional moduli of smoothness in Orlicz spaces
type: Item
instance of: scholarly article
title: Improved converse theorems and fractional moduli of smoothness in Orlicz spaces
zbMATH Open document ID: 1268.41013
author: Ramazan Akgün
published in: Bulletin of the Malaysian Mathematical Sciences Society. Second Series
publication date: 2013-03-11T00:00:00Z
full work available at URL: vol36 1 5.html
review text: Converse results of trigonometric approximation of functions and their fractional derivatives in certain Orlicz spaces are improved. More precisely, the converse inequality given by \textit{R. Akgün} and \textit{D. M. Israfilov} [Glas. Mat., III. Ser. 43, No. 1, 121--136 (2008; Zbl 1152.30038), Theroem 2] \[ \omega_{r}(f;1/(n+1))_{M}\leq \frac{C(M, r)}{(n+1)^{r}}\sum_{\mu=0}^{n}(\mu+1)^{r-1}E_{\mu}(f)_{M}, \] is improved to \[ \omega_{r}(f;1/(n+1))_{M}\leq \frac{C(M, r)}{(n+1)^{r}}\left \{\sum_{\mu=0}^{n}(\mu+1)^{rq-1}E_{\mu}^{q}(f)_{M}\right \}^{1/q}, \] for the case that \(M(u^{1/q})\) is convex for some \(1<q\leq 2\). For a definition of the appearing quantities in this estimate, see the paper.
Mathematics Subject Classification ID: 41A30
Mathematics Subject Classification ID: 42A10
Mathematics Subject Classification ID: 41A17
Mathematics Subject Classification ID: 41A27
zbMATH DE Number: 6143141
zbMATH Keywords: Orlicz space
zbMATH Keywords: trigonometric approximation
zbMATH Keywords: inverse approximation
zbMATH Keywords: fractional modulus of smoothness
reviewed by: Sorin G. Gal
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 6143141