Orthogonal polynomials and quadrature rules on the unit circle associated with perturbations of symmetric measures

JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS

卷:375

Article

Veronese, Daniel O.¹ 

[1] Univ Fed Triangulo Mineiro, Dept Matemat Aplicada, ICTE, BR-38064200 Uberaba, MG, Brazil

关键词: Quadrature rules;    Chain sequences;    Orthogonal polynomials on the unit circle;    Quasi-symmetric measures;   

DOI  :  10.1016/j.cam.2020.112808

来源: Elsevier

【 摘 要 】

It was shown recently that given a pair of real sequences {{c(n)}(n=1)(infinity), {dn}(n=1)(infinity)}, with {dn}(n=1)(infinity) a positive chain sequence, we can associate a unique nontrivial probability measure mu on the unit circle, and conversely. In this paper, we consider the set Q(c) of all nontrivial probability measures for which c(n) = (-1)(n)c, with c is an element of R. We show that every measure mu is an element of Q(c) can be obtained from a perturbation of a symmetric measure eta on [-1, 1]. Moreover, the sequence of orthogonal polynomials associated with mu can be given in terms of a perturbation of symmetric orthogonal polynomials associated with g. We also prove that every measure in Q(c) is quasi-symmetric, that is, there exists a complex valued function q((c))(z) satisfying d mu(z) = -q((c))(z)d mu((z) over bar), and such that g((c))(z) -> 1 when c -> 0. Quadrature rules with quasi-symmetric weights are also considered. Finally, some examples of orthogonal polynomials on the unit circle and its associated quasi-symmetric nontrivial probability measures are obtained. (C) 2020 Elsevier B.V. All rights reserved.

【 授权许可】

Free