On convergence and divergence of Fourier-Bessel series
type: Item
instance of: scholarly article
title: On convergence and divergence of Fourier-Bessel series
zbMATH Open document ID: 1042.42024
publication date: 2002-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/123691
full work available at URL: pp223-235.html
review text: In this interesting paper, the author investigates maximal inequalities for orthogonal expansions involving Bessel functions, and then deduces results on mean convergence, convergence almost everywhere, and divergence. Let \(\nu>-1\) and let \(J_\nu\) denote the Bessel function of the first kind of order \(\nu\). Let \(\{\lambda_n\}^\infty_{n= 1}\) denote its positive zeros in increasing order. There is the classical orthogonality result \[ \int^1_0 J_\nu(\lambda_n x)J_\nu(\lambda_m x)\,x\,dx= 0,\quad m\neq n. \] Form the orthonormal functions \[ \begin{aligned} \psi_n(x) &= c_n J_\nu(\lambda_n x)/x^\nu;\\ \phi_n(x) &= c_n J_\nu(\lambda_n x);\\ \varphi_n(x) &= c_n\sqrt{x} J_{n, \nu}(x),\end{aligned} \] \(n\geq 1\), where \(c_n\) is a positive normalizing constant. These are orthonormal, respectively, in \(L^2(0, 1)\) with weight \(x^{2\nu+1}\), \(L^2(0,1)\) with weight \(x\) and \(L^2(0,1)\) with weight 1. For a measurable function \(f\) on \((0, 1)\), form its orthonormal expansion in the \(\{\varphi_n\}\), \[ \sum^\infty_{n=1} a^\nu_n \varphi_n(x) \] and the partial sums \[ S_Nf(x)= \sum^N_{n=1} a^\nu_n \varphi_n(x). \] The author gives a new proof of a result of Guadalupe, Perez, Ruiz and Varona, that if \(\nu\geq -{1\over 2}\), \(1< p< \infty\), \(-1<\alpha< p-1\), \[ \biggl\|\sup_{N\geq 1}| S_N f|\biggl\|_{p,\alpha}\leq C\| f\|_{p,\alpha}, \] with \(C\) independent of \(f\), and with \[ \| f\|_{p,\alpha}= \Biggl(\int^1_0 | f(x)|^p x^\alpha \,dx\Biggr)^{1/p}. \] From this maximal function inequality, it follows that as \(N\to\infty\), \(S_N f(x)\to f(x)\) a.e. in \((0,1)\). The author proves similar results for orthonormal expansions in the \(\{\phi_n\}\) and \(\{\psi_n\}\). For example, if \(\nu> -1/2\), and \(4{v+1\over 2\nu+3}< p\leq\infty\), then the partial sums of the orthonormal expansions in \(\{\psi_n\}\) converge a.e. for each function \(g\) in \(L^p(x^{2n+1}dx, (0,1))\). The author shows the bound for \(p\) is sharp: for \(p= 4{v+1\over 2\nu+3}\), the…
Mathematics Subject Classification ID: 42C10
Mathematics Subject Classification ID: 33F99
zbMATH DE Number: 1893820
zbMATH Keywords: Fourier-Bessel expansion
zbMATH Keywords: orthonormal expansions
zbMATH Keywords: mean convergence
zbMATH Keywords: divergence
author: Krzysztof Stempak
reviewed by: Doron S. Lubinsky
MaRDI profile type: Publication
published in: ETNA - Electronic Transactions on Numerical Analysis
description: scientific article; zbMATH DE number 1893820