Expansions in Laguerre polynomials of negative order
type: Item
instance of: scholarly article
title: Expansions in Laguerre polynomials of negative order
zbMATH Open document ID: 0840.33004
published in: Journal of Approximation Theory
publication date: 1995-10-09T00:00:00Z
review text: Let \(n\) be a non-negative integer, \(-(n+2)< \alpha<- (n+1)\). Let \(f(x)\) be a given function, and suppose there exists a polynomial \(P_n (x)\) of degree \(n\) such that \[ \int_0^\infty |f(x)- P_n (x)|x^\alpha dx<\infty. \] The Hadamard's finite part (f.p.) of the infinite integral \(\int_0^\infty f(x) x^\alpha dx\) is defined by \[ \text{f.p. }\int_0^\infty f(x) x^\alpha dx= \int_0^\infty |f(x)- P_n(x) |x^\alpha dx. \] This paper is concerned with the Laguerre expansions of non-integer \(\alpha< -1\) defined by \[ f(x)\cong \sum^\infty_{k=0} a_k L_k^\alpha (x), \] where \(L_k^\alpha (x)\) is the \(k\)-th Laguerre polynomial of order \(\alpha\) and \[ a_k= {{k!} \over {\Gamma(k +\alpha+1)}} \text{ f.p. }\int_0^\infty f(y) L_k^\alpha (y) dy. \] The author proves the pointwise convergence of expansions of functions for which the difference of the function and a suitable polynomial satisfies a certain integrability condition.
Mathematics Subject Classification ID: 33C45
zbMATH DE Number: 803680
zbMATH Keywords: Laguerre polynomial
author: Elizabeth Kochneff
reviewed by: Luigi Gatteschi
MaRDI profile type: Publication
full work available at URL: jath.1995.1055
OpenAlex ID: W1980189716
DOI: 10.1006/JATH.1995.1055
description: scientific article; zbMATH DE number 803680