EXTREME ZEROS IN A SEQUENCE OF PARA-ORTHOGONAL POLYNOMIALS AND BOUNDS FOR THE SUPPORT OF THE MEASURE

Autor(es)

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

Afiliação do(s) autor(es)

^[1] Univ Almeria, Dept Matemat, Almeria 04120 - Spain

^[2] Granada Univ, Inst Carlos Fis Teor & Computac 1, Granada - Spain

^[3] Univ Estadual Paulista, UNESP, IBILCE, Dept Matemat Aplicada, BR-15054000 Sao Jose Do Rio Preto, SP - Brazil

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

• Tipo de documento: Artigo Científico
• Fonte: Mathematics of Computation; v. 87, n. 309, p. 261-288, JAN 2018.
• Citações Web of Science: 3

Resumo

Given a nontrivial Borel measure mu on the unit circle T, the corresponding reproducing (or Christoffel-Darboux) kernels with one of the variables fixed at z = 1 constitute a family of so-called para-orthogonal polynomials, whose zeros belong to T. With a proper normalization they satisfy a three-term recurrence relation determined by two sequences of real coefficients, [c(n)] and [d(n)], where [d(n)] is additionally a positive chain sequence. Coefficients (c(n), d(n)) provide a parametrization of a family of measures related to mu by addition of a mass point at z = 1. In this paper we estimate the location of the extreme zeros (those closest to z = 1) of the para-orthogonal polynomials from the (c(n), d(n))-parametrization of the measure, and use this information to establish sufficient conditions for the existence of a gap in the support of mu at z = 1. These results are easily reformulated in order to find gaps in the support of mu at any other z epsilon T. We provide also some examples showing that the bounds are tight and illustrate their computational applications. (AU)

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