Christoffel formula for kernel polynomials on the unit circle

Autor(es)

Bracciali, C. F. ^[1] ; Martinez-Finkelshtein, A. ^{[2, 3]} ; Sri Ranga, A. ^[1] ; Veronese, D. O. ^[4]

Afiliação do(s) autor(es)

^[1] Univ Estadual Paulista, UNESP, IBILCE, DMAp, BR-15054000 Sao Jose Do Rio Preto, SP - Brazil

^[2] Univ Almeria, Dept Matemat, Almeria 04120 - Spain

^[3] Granada Univ, Inst Carlos Fis Teor & Computac, Granada - Spain

^[4] Univ Fed Triangulo Mineiro, ICTE, BR-38064200 Uberaba, MG - Brazil

• Tipo de documento: Artigo Científico
• Fonte: Journal of Approximation Theory; v. 235, p. 46-73, NOV 2018.
• Citações Web of Science: 1

Resumo

Given a nontrivial positive measure it on the unit circle T, the associated Christoffel-Darboux kernels are K-n(z, w; mu) = Sigma(n)(k=0)<(phi(k)(w;mu))over bar>phi(k)(z; mu), n > 0, where phi(k)(.; mu) are the orthonormal polynomials with respect to the measure mu. Let the positive measure nu on the unit circle be given by d nu(z) = vertical bar G(2m) (z)vertical bar d mu(z), where G(2m) is a conjugate reciprocal polynomial of exact degree 2m. We establish a determinantal formula expressing [K-n(z, w; nu)]n >= 0 directly in terms of [K-n(z, w; mu)]n >= 0. Furthermore, we consider the special case of w = 1; it is known that appropriately normalized polynomials K-n(z, 1; mu) satisfy a recurrence relation whose coefficients are given in terms of two sets of real parameters [c(n)(mu)](n=1)(infinity) and [g(n)(mu)(n=1)(infinity), with 0 < g(n) < 1 for n >= 1. The double sequence [c(n)(mu), g(n)(mu))](n=1)(infinity) characterizes the measure mu. A natural question about the relation between the parameters c(n) (mu), g(n)(mu), associated with mu, and the sequences c(n)(nu), g(n)(nu), corresponding to nu, is also addressed. Finally, examples are considered, such as the Geronimus weight (a measure supported on an arc of T), a measure for which the Christoffel-Darboux kernels, with w = 1, are given by basic hypergeometric polynomials and a measure for which the orthogonal polynomials and the Christoffel-Darboux kernels, again with w = 1, are given by hypergeometric polynomials. (C) 2018 Elsevier Inc. All rights reserved. (AU)

• Processo FAPESP: 17/12324-6 - Polinômios ortogonais no círculo unitário e estudos relacionados
• Beneficiário: Alagacone Sri Ranga
• Modalidade de apoio: Auxílio à Pesquisa - Regular
• Processo FAPESP: 16/09906-0 - Análise harmônica, teoria da aproximação, funções especiais e aplicações
• Beneficiário: Dimitar Kolev Dimitrov
• Modalidade de apoio: Auxílio à Pesquisa - Temático