Lagrange interpolation in the zeros of Bessel functions by entire functions of exponential type and mean convergence
type: Item
instance of: scholarly article
title: Lagrange interpolation in the zeros of Bessel functions by entire functions of exponential type and mean convergence
zbMATH Open document ID: 0858.30022
published in: Methods and Applications of Analysis
publication date: 1997-03-31T00:00:00Z
review text: Let \(J_\alpha\) be the Bessel function of the first kind of order \(\alpha>-1\); \(G_a:=z^{-\alpha}J_\alpha(z)\); \[ L_{\tau,\alpha}(f,z):=\sum^\infty_{\nu=-\infty,\nu\neq 0} {G_\alpha(\tau z)\over G_\alpha'(j_\nu)(\tau z-j_\nu)} f\Biggl({j_\nu\over \tau}\Biggr), \] where \(f:\mathbb{R}\to \mathbb{C}\), \(\tau>0\) is non-integer, \(j_\kappa\) is the \(\kappa\)th positive zero of \(J_\alpha\), \(j_{-\kappa}=-j_\kappa\), \(\kappa\in \mathbb{N}\). The authors had studied the mean convergence of \(L_{\tau,\alpha}(f,.)\) to \(f\) as \(\tau\to\infty\) for a sufficient wide subclass of Riemann integrable functions. This main result is analogous to two well-known theorems of J. Marcinkiewicz and R. Askey. For the proof of it many lemmas were needed. Some of them are of independent interest.
Mathematics Subject Classification ID: 30D10
Mathematics Subject Classification ID: 33C10
Mathematics Subject Classification ID: 41A05
zbMATH DE Number: 935320
zbMATH Keywords: entire functions of exponential type
zbMATH Keywords: mean convergence
zbMATH Keywords: Bessel function
author: Georgi R. Grozev
author: Qazi I. Rahman
reviewed by: P. Z. Agranovich
MaRDI profile type: Publication
full work available at URL: maa.1996.v3.n1.a4
OpenAlex ID: W311970520
DOI: 10.4310/MAA.1996.V3.N1.A4
description: scientific article; zbMATH DE number 935320