Division problem of moment functionals.
type: Item
instance of: scholarly article
title: Division problem of moment functionals.
zbMATH Open document ID: 1043.42016
author: D. Massart
published in: Rocky Mountain Journal of Mathematics
publication date: 2003-12-03T00:00:00Z
full work available at URL: CONT32-2.html
review text: The authors study moment generating functionals \(\sigma\), i.e., linear functions on the space \textbf{P} of all polynomials, satisfying the Hamburger condition \[ \Delta:=\text{det}\,\left( \sigma_{i+j}\right)_{i,j,=0}^n \not= 0,\;n\geq 0, \] with \(\sigma_n:=\langle \sigma, x^n\rangle,\;n\geq 0\). The paper gives necessary and sufficient conditions for a moment functional \(\tau\) to be also quasi-definite, where \(\tau\) is connected with \(\sigma\) by \[ D(x)\tau = A(x)\sigma, \] where \(A(x),\,D(x)\) are co-prime, non-zero polynomials. When \(\tau\) is quasi-definite, they exhibit a rather simple representation of the monic orthogonal polynomials with respect to \(\tau\) in terms of those orthogonal relative to \(\sigma\). As corollaries some previous results are recovered, due to \textit{S. Belmehdi} [IMACS Ann. Comput. Appl. Math. 9, 169--176 (1991; Zbl 0837.42013)] and \textit{A. Branquinho} and \textit{F. Marcellán} [Int. J. Math. Math. Sci. 19, No. 4 , 643--656 (1996; Zbl 0846.33008)]. The final section of the paper treats two examples: \[ x^3\tau=\sigma,\;\sigma\text{ the Laguerre moment functional on }(0,\infty), \] \[ (1+x)\tau=(1-x)\sigma,\;\sigma\text{ the Jacobi moment functional on }(-1,1). \]
Mathematics Subject Classification ID: 42C05
Mathematics Subject Classification ID: 33C47
zbMATH DE Number: 2012116
zbMATH Keywords: quasi-definite moment functionals
zbMATH Keywords: division problem
reviewed by: Marcel G. de Bruin
MaRDI profile type: Publication
OpenAlex ID: W1988438756
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DOI: 10.1216/RMJM/1030539695
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description: scientific article; zbMATH DE number 2012116