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On Finite-Term Recurrence Relations for Bergman and Szegő Polynomials

On Finite-Term Recurrence Relations for Bergman and Szegő Polynomials With the aid of Havin’s Lemma (which we generalize) we prove that polynomials orthogonal over the unit disk with respect to certain weighted area measures (Bergman polynomials) cannot satisfy a finite-term recurrence relation unless the weight is radial, in which case the polynomials are simply monomials. For polynomials orthogonal over the unit circle (Szegő polynomials) we provide a simple argument to show that the existence of a finite-term recurrence implies that the weight must be the reciprocal of the square modulus of a polynomial. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Computational Methods and Function Theory Springer Journals

On Finite-Term Recurrence Relations for Bergman and Szegő Polynomials

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References (16)

Publisher
Springer Journals
Copyright
Copyright © 2012 by Heldermann  Verlag
Subject
Mathematics; Analysis; Computational Mathematics and Numerical Analysis; Functions of a Complex Variable
ISSN
1617-9447
eISSN
2195-3724
DOI
10.1007/BF03321834
Publisher site
See Article on Publisher Site

Abstract

With the aid of Havin’s Lemma (which we generalize) we prove that polynomials orthogonal over the unit disk with respect to certain weighted area measures (Bergman polynomials) cannot satisfy a finite-term recurrence relation unless the weight is radial, in which case the polynomials are simply monomials. For polynomials orthogonal over the unit circle (Szegő polynomials) we provide a simple argument to show that the existence of a finite-term recurrence implies that the weight must be the reciprocal of the square modulus of a polynomial.

Journal

Computational Methods and Function TheorySpringer Journals

Published: Mar 21, 2012

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