Orthogonal Polynomials, Verblunsky Coefficients, and a Szegő-Verblunsky Theorem on the Unit Sphere in $\mathbb{C}^d$

Connor J. Gauntlett; David P. Kimsey

Paper

Orthogonal Polynomials, Verblunsky Coefficients, and a Szegő-Verblunsky Theorem on the Unit Sphere in $\mathbb{C}^d$

Abstract

Given a measure

on the unit sphere

with Lebesgue decomposition

, with respect to the rotation-invariant Lebesgue measure

, we introduce notions of orthogonal polynomials

, Verblunsky coefficients

, and an associated Christoffel function

, and we prove a recurrence relation for the orthogonal polynomials involving the Verblunsky coefficients reminiscent of the classical Szegő recurrences, as well as an analogue of Verblunsky's theorem. Moreover, we establish a number of equalities involving the orthogonal polynomials, determinants of moment matrices, and the Christoffel function, and show that if

is discrete, then the aforementioned quantities depend only on the absolutely continuous part of

. If, in addition to

being discrete, one is able to find

such that

and

then we establish a

-variate Szegő-Verblunsky theorem, namely

Finally, we identify several classes of weights where one may construct such an

and highlight an explicit example of a weight

, residing outside of these classes, where