On the norms of generalized translation operators generated by the Jacobi-Dunkl operators
type: Item
instance of: scholarly article
title: On the norms of generalized translation operators generated by the Jacobi-Dunkl operators
zbMATH Open document ID: 1266.47070
author: O. L. Vinogradov
published in: Journal of Mathematical Sciences (New York)
publication date: 2012-11-28T00:00:00Z
review text: The author studies operators of generalized translation generated by the Dunkl type operators \(\Lambda\) of the form \[ T_\Lambda^y\,f(x)=\int_{-\pi}^\pi f(z)W(x,y,z)A_{\alpha,\beta}(z)\,dz \] in the spaces \(L^p_{\alpha,\beta}[-\pi,\pi]\) with the weight \[ A_{\alpha,\beta}(x)=(1-\cos x)^\alpha (1+\cos x)^\beta |\sin x|. \] Here, \(W\) is a specific kernel which involves only sine and cosine functions of \(x,y,z\). One of the main results provides a bound of the norm of such an operator for \(1\leq p\leq+\infty\) of the form \(\log \|T_\Lambda^y\|\leq |1-2/p|\log 2\).
Mathematics Subject Classification ID: 47G10
Mathematics Subject Classification ID: 47A30
zbMATH DE Number: 6110762
zbMATH Keywords: generalized translation operators
zbMATH Keywords: Jacobi-Dunkl weights
zbMATH Keywords: integral operators
zbMATH Keywords: norm
reviewed by: Leonid Golinskii
MaRDI profile type: Publication
full work available at URL: s10958-012-0765-8
OpenAlex ID: W1979833096
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DOI: 10.1007/S10958-012-0765-8
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description: scientific article; zbMATH DE number 6110762