Spectral measures corresponding to orthogonal polynomials with unbounded recurrence coefficients
type: Item
instance of: scholarly article
title: Spectral measures corresponding to orthogonal polynomials with unbounded recurrence coefficients
zbMATH Open document ID: 0678.47017
DOI: 10.1007/BF01889616
author: S. H. Smith
published in: Constructive Approximation
publication date: 1989-01-01T00:00:00Z
review text: The author studies sequences of polynomials that satisfy recursion formulas of the form \[ P_1(\lambda)=1, \quad P_2(\lambda) = (\lambda-b_1)/a_1, \quad P_ n(\lambda) = [(\lambda - b_{n-1}) P_{n-1} (\lambda) - a_{n-2} P_{n-2} (\lambda)] /a_{n-1}, \quad n=3,4,\ldots, \] with \(a_ n\) and \(b_ n\) real and \(a_ n>0\). It is known that if \(\{a_ n\}\) and \(\{b_ n\}\) are bounded or if lim \(a_ n=\infty\) and \(\sum^{\infty}_{n=1}1/a_ n=\infty\), then there exists a unique measure \(\mu\), supported on the real line, with respect to which the polynomials are orthogonal. Properties of \(\mu\) have been studied in the past by various authors. In the present paper the author continues this study for the case that lim \(a_ n=\infty\), \(\sum^{\infty}_{n=1}1/a_ n=\infty\), and lim \(b_ n=b\). The plan of the paper is to associate with the system of polynomials a tridiagonal matrix which represents an unbounded selfadjoint operator C in \(\ell^ 2\). C is unitarily equivalent to the multiplication oprator in \(L^ 2(\mu)\), where \(\mu\) is the measure mentioned above. Additional conditions are imposed on \(\{a_ n\}\) and \(\{b_ n\}\) which provide information about the spectral measure E(\(\lambda)\) of C and hence about \(\mu\). Following is an example of the type of result that is proved: Let \(d_ n=| a_ n-a_{n-1}|\). Suppose that \(\sum^{\infty}_{n=2}[a^ 2_ n-a^ 2_{n-1}]^- < \infty\), \(\sum^\infty_{n=2} | d_ n-d_{n-1}| < \infty,\) and \(\sum^\infty_{n=2} a_{n-1} | b_ n-b_{n-1}| < \infty\). (Here, \([x]^- = ([x]-x)/2\).) Then, \(\mu\) has an absolutely continuous part with support sp(C). The case \(b_ n=0\) has been previously studied by the author [Proc. Am. Math. Soc. 100, No.3, 457-463 (1987); SIAM J. Math. Anal. 19, No.4, 939-943 (1988)].
Mathematics Subject Classification ID: 47B15
Mathematics Subject Classification ID: 47B37
Mathematics Subject Classification ID: 47B39
Mathematics Subject Classification ID: 47A66
Mathematics Subject Classification ID: 42C05
Mathematics Subject Classification ID: 33C45
zbMATH DE Number: 4111372
zbMATH Keywords: orthogonal polynomial
zbMATH Keywords: spectral measure
zbMATH Keywords: absolutely continuous measure
zbMATH Keywords: recursion formulas
zbMATH Keywords: tridiagonal matrix
zbMATH Keywords: unbounded selfadjoint operator
zbMATH Keywords: unitarily equivalent to the multiplication oprator
reviewed by: Richard C. Gilbert
MaRDI profile type: Publication
cites work: Q4720165
cites work: Tridiagonal matrix representations of cyclic self-adjoint operators. II
cites work: Cyclic Operators, Commutators, and Absolutely Continuous Measures
cites work: Spectral Measures, Orthogonal Polynomials, and Absolute Continuity
cites work: Q5336643
cites work: Perturbation of continuous spectra by Trace class operators
cites work: Q3327119
cites work: Orthogonal polynomials
cites work: Q5528957
cites work: Q5845997
cites work: Q4334625
full work available at URL: https://doi.org/10.1007/bf01889616
OpenAlex ID: W1992588234
description: scientific article; zbMATH DE number 4111372