Weighted norm inequalities for fractional integrals with an application to mean convergence of Laguerre series
type: Item
instance of: scholarly article
title: Weighted norm inequalities for fractional integrals with an application to mean convergence of Laguerre series
zbMATH Open document ID: 0887.26005
published in: Journal of Approximation Theory
publication date: 1998-01-11T00:00:00Z
review text: The paper deals with the weighted norm inequalities in \(L_p\)-spaces for the Riemann-Liouville fractional integral \[ f_a(x)= {1\over\Gamma(a)}\int^x_0 {f(t)dt\over(x- t)^{1-a}} (a>0) \] [see, for example, Section 5 in the book by \textit{S. G. Samko}, \textit{A. A. Kilbas} and \textit{O. I. Marichev}: ``Fractional integrals and derivatives. Theory and Applications'' (Russian, 1987; Zbl 0617.26004, English translation, 1993; Zbl 0818.26003)]. Sufficient conditions are given for the validity of the inequality with power exponential weights \[ \int^\infty_0|x^{-(m+ l)}e^{-bx} f_a(x)|^q dx\leq K\Biggl(\int^\infty_0 |x^{-l} e^{-bx}f(x)|^p dx\Biggr)^{q/p} \] with \(p>1\), \(b>0\), \(m\geq 0\), \(l>1/p- 1\) and \(q\geq p\). Examples are given to show the accuracy of the conditions on parameters. The obtained results are applied on certain functions \(f(x)\) to show that their expansions with respect to Laguerre polynomials \(L^\lambda_k(x)\) are convergent in \(L_p\)-norm. It is proved the existence of constants \(c_k\) \((k= 0,1,2,\dots)\) so that \[ \Biggl|e^{-x/2} x^{\lambda/2}\Biggl( \sum^n_{k= 0} c_kL^\lambda_k- f\Biggr)\Biggr|_q\to 0\quad\text{as }n\to\infty. \] Such a relation for the standard range of \(q\in(4/3,4)\) was obtained by \textit{R. Askey} and \textit{S. Wainger} [Am. J. Math. 87, 695-708 (1965; Zbl 0125.31301)]. The author extends this result to some \(q\) outside of \((4/3,4)\). The example \(f(x)= x^\nu\) illustrates the above mean convergence of the Laguerre expansion.
Mathematics Subject Classification ID: 26A33
Mathematics Subject Classification ID: 41A58
Mathematics Subject Classification ID: 26D10
zbMATH DE Number: 1076154
zbMATH Keywords: Laguerre series
zbMATH Keywords: weighted norm inequalities
zbMATH Keywords: Riemann-Liouville fractional integral
zbMATH Keywords: power exponential weights
zbMATH Keywords: Laguerre expansion
author: Elizabeth Kochneff
reviewed by: A. A. Kilbas
MaRDI profile type: Publication
full work available at URL: jath.1996.3089
OpenAlex ID: W2036189690
cites work: Mean Convergence of Expansions in Laguerre and Hermite Series
cites work: Q5827353
cites work: Q5812325
cites work: On Certain Convolution Inequalities
cites work: The special functions and their approximations. Vol. I, II
cites work: L'intégrale de Riemann-Liouville et le problème de Cauchy
cites work: A Characterization of Two Weight Norm Inequalities for Fractional and Poisson Integrals
cites work: Weighted Norm Inequalities for Fractional Integrals
DOI: 10.1006/JATH.1996.3089
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description: scientific article; zbMATH DE number 1076154