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Para-orthogonal polynomials on the unit circle satisfying three term recurrence formulas

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Autor(es):
Bracciali, C. F. ; Ranga, A. Sri ; Swaminathan, A.
Número total de Autores: 3
Tipo de documento: Artigo Científico
Fonte: APPLIED NUMERICAL MATHEMATICS; v. 109, p. 19-40, NOV 2016.
Citações Web of Science: 6
Resumo

When a nontrivial measure mu on the unit circle satisfies the symmetry d mu(e(i(2 pi-theta))) = -d mu(e(i theta)) then the associated orthogonal polynomials on the unit circle, say Phi(n), are all real. In this case, in 1986, Delsarte and Genin have shown that the two sequences of para-orthogonal polynomials [z Phi(n)(z) + Phi{*}(n)(z)] and [z Phi(n)(z) - Phi{*}(n)(z)], where Phi{*}(n)(z) = z(n)<(Phi(n)(1 root z))over bar> satisfy three term recurrence formulas and have also explored some further consequences of these sequences of polynomials such as their connections to sequences of orthogonal polynomials on the interval {[}-1, 1]. The same authors, in 1988, have also provided a means to extend these results to cover any nontrivial measure on the unit circle. However, only recently the extension associated with the para-orthogonal polynomials z Phi(n)(z) - Phi{*}(n)(z) was thoroughly explored, especially from the point of view of three term recurrence and chain sequences. The main objective of the present article is to provide the theory surrounding the extension associated with the para-orthogonal polynomials z Phi(n)(z) + Phi{*}(n)(z) for any nontrivial measure on the unit circle. As an important application of the theory, a characterization for the existence of the integral integral(2 pi)(0) vertical bar e(i theta) - w vertical bar(-2)d mu(e(i theta)) where w is such that vertical bar w vertical bar = 1, is given in terms of the coefficients alpha(n-1) = -<(Phi(n)(0))over bar>, n >= 1. Examples are also provided to justify all the results. (C) 2016 IMACS. Published by Elsevier B.V. All rights reserved. (AU)

Processo FAPESP: 09/13832-9 - Polinômios ortogonais, funções especiais e aplicações
Beneficiário:Dimitar Kolev Dimitrov
Linha de fomento: Auxílio à Pesquisa - Temático