Nonclassical Jacobi polynomials and Sobolev orthogonality
type: Item
instance of: scholarly article
title: Nonclassical Jacobi polynomials and Sobolev orthogonality
zbMATH Open document ID: 1259.33015
author: Andrea Bruder
author: Lance L. Littlejohn
published in: Results in Mathematics
publication date: 2013-01-10T00:00:00Z
review text: The authors consider the nonclassical Jacobi polynomials with parameters \(a>-1\) and \(b=-1\). They prove that the sequence of Jacobi polynomials \(\{{P_{n}^{(a,-1)}}\}_{n=0}^{\infty}\) is complete in the Hilbert-Sobolev space \(W_{a}\) with the inner product \[ \phi (f,g)=f(-1){\bar{g}}(-1)+\int_{-1}^{1}f^{\prime}(t){\bar{g}}^{\prime}(t)(1-t)^{a+1}\,dt. \] They also construct a self-adjoint positively bounded from below operator \(T_{a}\) in \(W_{a}\) having the Jacobi polynomials \(\{{P_{n}^{(a,-1)}}\}_{n=0}^{\infty}\) as a complete set of eigenfunctions.
Mathematics Subject Classification ID: 33C45
Mathematics Subject Classification ID: 33B30
Mathematics Subject Classification ID: 47B25
Mathematics Subject Classification ID: 34B20
Mathematics Subject Classification ID: 47B65
zbMATH DE Number: 6124391
zbMATH Keywords: nonclassical Jacobi polynomials
zbMATH Keywords: Sobolev orthogonality
reviewed by: Chrysoula G. Kokologiannaki
MaRDI profile type: Publication
full work available at URL: s00025-011-0102-4
OpenAlex ID: W2160013479
cites work: Q4877167
cites work: Orthogonality of the Jacobi polynomials with negative integer parameters
cites work: Sobolev orthogonal polynomials: The discrete-continuous case
cites work: Sobolev orthogonality for the Gegenbauer polynomials \(\{ C_n^{-N+1/2}\}_{n\geq 0}\)
cites work: An integral operator inequality with applications
cites work: Q4170993
cites work: Q5336643
cites work: Jacobi-Stirling numbers, Jacobi polynomials, and the left-definite analysis of the classical Jacobi differential expression
cites work: The Sobolev orthogonality and spectral analysis of the Laguerre polynomials \(\{L_{n}^{-k}\}\) for positive integers \(k\)
cites work: Q4882027
cites work: Sobolev orthogonal polynomials and second-order differential equations
cites work: A general left-definite theory for certain self-adjoint operators with applications to differential equations
cites work: Q5634277
cites work: Q3267566
DOI: 10.1007/S00025-011-0102-4
description: scientific article; zbMATH DE number 6124391