A Favard type theorem for orthogonal polynomials on the unit circle from a three term recurrence formula

TL;DR

This paper establishes a Favard type theorem linking a specific three-term recurrence for polynomials to a unique probability measure on the unit circle, revealing how chain sequence parameters influence the measure and orthogonal polynomials.

Contribution

It introduces a new Favard type theorem for orthogonal polynomials on the unit circle derived from a three-term recurrence involving chain sequences, detailing the measure's dependence on sequence parameters.

Findings

Existence of a unique probability measure on the unit circle for the given recurrence.

The measure's mass at z=1 is determined by the maximal parameter sequence.

The chain sequence parameter d_1 influences the measure but not the polynomials.

Abstract

The objective of this manuscript is to study directly the Favard type theorem associated with the three term recurrence formula % \[ R_{n+1}(z) = \big[(1+ic_{n+1})z+(1-ic_{n+1})\big] R_{n}(z) - 4 d_{n+1} z R_{n-1}(z), \quad n \geq 1, \] % with $R_{0}(z) =1$ and $R_{1}(z) = (1+ic_{1})z+(1-ic_{1})$, where $\{c_n\}_{n=1}^{\infty}$ is a real sequence and $\{d_n\}_{n=1}^{\infty}$ is a positive chain sequence. We establish that there exists an unique nontrivial probability measure $\mu$ on the unit circle for which $\{R_n(z) - 2(1-m_n)R_{n-1}(z)\}$ gives the sequence of orthogonal polynomials. Here, $\{m_n\}_{n=0}^{\infty}$ is the minimal parameter sequence of the positive chain sequence $\{d_n\}_{n=1}^{\infty}$. The element $d_1$ of the chain sequence, which does not effect the polynomials $R_n$, has an influence in the derived probability measure $\mu$ and hence, in the associated orthogonal polynomials on the unit circle. To be precise, if $\{M_n\}_{n=0}^{\infty}$ is the maximal parameter sequence of the chain sequence, then the measure $\mu$ is such that $M_0$ is the size of its mass at $z=1$. An example is also provided to completely illustrates the results obtained.