Properties of the operator domains of the fourth-order Legendre-type differential expressions
type: Item
instance of: scholarly article
title: Properties of the operator domains of the fourth-order Legendre-type differential expressions
zbMATH Open document ID: 0803.34014
author: S. M. Loveland
author: W. N. Everitt
author: Lance L. Littlejohn
published in: Differential and Integral Equations
publication date: 1995-01-03T00:00:00Z
review text: Legendre-type polynomials are obtained when the Gram-Schmidt process is applied to \(\{x^ n: n\in \mathbb{N}_ 0\}\) in \(L^ 2([-1,1]; \mu)\), where the measure \(\mu\) is generated by \[ \widehat\mu(x)= \begin{cases} -1-M, & x\in (-\infty,-1],\\ x, & x\in (-1,1),\\ -1+ N, & x\in [1,\infty),\end{cases} \] and \(M\), \(N\) are non-negative constants. In each of the five possible cases the polynomials satisfy a formally symmetric spectral differential equation: one of these is of second order (the classical case \(M= N=0\)), three of fourth order (discovered by H. L. Krall) and one of sixth order (discovered by L. L. Littlejohn). This paper deals with the three cases for which the differential equations are of fourth order. The fourth order differential expressions generate self-adjoint operators in \(L^ 2([-1,1]; \mu)\) with discrete spectra and whose eigenfunctions are the appropriate Legendre-type polynomials. Properties of the domains of these operators are determined.
Mathematics Subject Classification ID: 34B20
Mathematics Subject Classification ID: 33C65
Mathematics Subject Classification ID: 41A10
Mathematics Subject Classification ID: 47E05
Mathematics Subject Classification ID: 34L05
zbMATH DE Number: 597490
zbMATH Keywords: Legendre-type polynomials
zbMATH Keywords: Gram-Schmidt process
zbMATH Keywords: formally symmetric spectral differential equation
zbMATH Keywords: fourth order differential expressions
zbMATH Keywords: eigenfunctions
reviewed by: W. D. Evans
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 597490