ANÁLISE DE UM TEOREMA DO TIPO FAVARD NO CÍRCULO UNITÁRIO: POLINÔMIOS ORTOGONAIS E RELAÇÕES DE RECORRÊNCIA

Abstract

Favard's Theorem, widely known in the context of the real line, ensures the existence of a unique probability measure for which certain polynomials, satisfying a three term recurrence relation, are orthogonal. On the unit circle, there is a well established version of this result. However, unlike the real line case, the measure obtained in this configuration does not generate orthogonal polynomials that satisfy a three term recurrence relation. This article, based on a Master's research in Mathematics conducted at the Federal University of Maranhão, investigates a Favard type Theorem on the unit circle, as presented by (Castillo-Costa-Ranga-Veronese, 2014). The study focuses on polynomials that satisfy a three term recurrence relation, where the coefficients appearing in this relation are real sequences, including a positive chain sequence. The main objective is to explore and detail the technical aspects that support this theorem, filling gaps and complementing results not explained in the original work.

Keywords: Orthogonal polynomials on the unit circle. Three term recurrence formula. Positive chain sequences. Continued fractions.