Fourier-Bessel-type series: the fourth-order case
type: Item
instance of: scholarly article
title: Fourier-Bessel-type series: the fourth-order case
zbMATH Open document ID: 1183.33008
published in: Journal of Approximation Theory
publication date: 2009-12-03T00:00:00Z
review text: The structured higher-order Bessel-type functions were introduced in 1994. They satisfy linear ordinary differential equations with analytic coefficients in the complex plane \(\mathbb{C}\), the so-called higher-order Bessel-type differential equations. There is a countable infinity of these equations, all of them of even order. These differential equations possess many of the properties of the classical second-order Bessel differential equation, but these higher-order cases also bring remarkable new analytic structures. In many ways it is sufficient to study the properties of the fourth-order Bessel-type differential equation to be able to assess the corresponding properties of the sixth- and higher-order cases. This paper follows a number of earlier papers devoted to the study of the fourth-order case. On the positive half \([ 0,+ \infty )\) of the real line this differential equation has the Lagrange symmetric form \(L_M [ y ] (x) := \bigl( x y''(x) \bigr)'' - \bigl( ( 9x^{-1} + 8 M^{-1} x) y'(x) \bigr)' = \Lambda \;x \;y(x)\) for all \(x \in (0,\infty)\). Here \(M \in (0,\infty)\) is a given parameter and \(\Lambda \in \mathbb{C}\) is the spectral parameter. The spectral theory of this differential equation has been considered in certain weighted Lebesgue and Lebesgue-Stieltjes Hilbert function spaces. In this paper the spectral theory of the fourth-order Bessel-type differential equation is developed in the Lebesgue-Stieltjes space \(L^2([ 0,1 ] ; m_k )\) consisting of all Borel measurable functions \(f: [ 0,1 ] \rightarrow \mathbb{C}\) for which the norm and inner-product are given by \(\| f\|_k^2 = k \;|f(0)|^2 + \int_0^1 x \;|f(x)|^2 \;dx\) and \((f,g)_k = k f(0) \;\overline{g(0)} + \int_0^1 x \;f(x) \;\overline{g(x)} \;dx\) with \(k \in (0,\infty)\) a parameter. It is shown that for all values of the two parameters \(k,M \in (0,\infty)\) the self-adjoint operators generated by the Lagrange symmetric differential expression \(L_M\) in the space \(L^2([ 0,1 ]…
reviewed by: Nele De Schepper
Mathematics Subject Classification ID: 33C10
Mathematics Subject Classification ID: 34L10
Mathematics Subject Classification ID: 26A03
zbMATH DE Number: 5641721
zbMATH Keywords: Fourier-Bessel series
zbMATH Keywords: Bessel functions
zbMATH Keywords: Bessel-type functions
zbMATH Keywords: Dini boundary conditions
zbMATH Keywords: zeros of Bessel functions
author: W. N. Everitt
author: Clemens Markett
MaRDI profile type: Publication
full work available at URL: j.jat.2007.11.004
OpenAlex ID: W2022133949
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DOI: 10.1016/J.JAT.2007.11.004
description: scientific article; zbMATH DE number 5641721