Christoffel formula for kernel polynomials on the unit circle

Author(s)

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

Affiliation

^[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

• Document type: Journal article
• Source: Journal of Approximation Theory; v. 235, p. 46-73, NOV 2018.
• Web of Science Citations: 1

Abstract

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)

• FAPESP's process: 17/12324-6 - Orthogonal polynomials on the unit circle and related studies
• Grantee: Alagacone Sri Ranga
• Support Opportunities: Regular Research Grants
• FAPESP's process: 16/09906-0 - Harmonic analysis, approximation theory and applications
• Grantee: Dimitar Kolev Dimitrov
• Support Opportunities: Research Projects - Thematic Grants