Fourier-Bessel series for second-order and fourth-order Bessel differential equations
type: Item
instance of: scholarly article
title: Fourier-Bessel series for second-order and fourth-order Bessel differential equations
zbMATH Open document ID: 1152.34062
DOI: 10.1007/BF03321704
author: W. N. Everitt
author: Clemens Markett
published in: Computational Methods and Function Theory
publication date: 2008-07-29T00:00:00Z
review text: The authors consider the Hilbert function space framework for Fourier-Bessel series, based on linear differential operators generated by the second-order Bessel differential equation for Bessel functions of order zero \(-(xy'(x))'=\lambda x y(x)\), \(x\in(0,1]\), \(\lambda\in\mathbb C\) [see \textit{M. A. Naimark}, Linear differential operators. Part II, New York: Frederick Ungar Publishing Co. (1968; Zbl 0227.34020)] and by the fourth-order Bessel-type differential equation \[ (xy''(x))''-((9x^{-1}+8M^{-1}x)y'(x))'=\Lambda x y(x),\quad x \in(0,1],\tag{*} \] where \(M>0\), \(\lambda\in\mathbb C\) is the spectral parameter. In both cases the analysis is concerned with the theory of unbounded linear differential operators in the Hilbert function space \(L^2((0,1);x)\). It is proved that for \(f\in L^2((0,1);x)\) the Fourier-Bessel series \(f(x)=\sum_{m=1}^\infty a_mJ_0(j_mx)\), where \(j_m\) are positive zeros of \(J_0(x)\) and \(a_m=2J^{-2}_1(j_m) \int_0^1tf(t)J_0(j_mt)\,dt\) converges to \(f\) in \(L^2((0,1);x)\). A similar result is obtained for solutions to the fourth-order Bessel-type differential equation (*).
reviewed by: Valery V. Karachik
Mathematics Subject Classification ID: 34L10
Mathematics Subject Classification ID: 33C10
Mathematics Subject Classification ID: 33C05
zbMATH DE Number: 5305554
zbMATH Keywords: Fourier-Bessel series
zbMATH Keywords: Bessel functions
zbMATH Keywords: Bessel-type functions
zbMATH Keywords: eigenfunction expansions
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 5305554