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

Abstract

Given a measure $\mu$ on the unit sphere $\partial\mathbb{B}^d$ in $\mathbb{C}^d$ with Lebesgue decomposition ${\rm d} \mu = w \, {\rm d} \sigma + {\rm d} \mu_s$, with respect to the rotation-invariant Lebesgue measure $\sigma$ on $\partial \mathbb{B}^d$, we introduce notions of orthogonal polynomials $(\varphi_{\alpha})_{\alpha \in \mathbb{N}_0^d}$, Verblunsky coefficients $(\gamma_{\alpha,\beta})_{\alpha,\beta \in \mathbb{N}_0^d}$, and an associated Christoffel function $\lambda_{\infty}^{(d)}(z; {\rm d} \mu)$, and we prove a recurrence relation for the orthogonal polynomials involving the Verblunsky coefficients reminiscent of the classical Szeg\H{o} 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 ${\rm supp}\, \mu_s$ is discrete, then the aforementioned quantities depend only on the absolutely continuous part of $\mu$. If, in addition to ${\rm supp}\, \mu_s$ being discrete, one is able to find $f \in H^{\infty}(\mathbb{B}^d)$ such that $f(0) = 1$ and $$\int_{\partial \mathbb{B}^d} |f(\zeta)|^2 w(\zeta) {\rm d}\sigma(\zeta) \leq \exp\left( \int_{\partial \mathbb{B}^d} \log(w(\zeta)) \, {\rm d}\sigma(\zeta) \right),$$ then we establish a $d$-variate Szeg\H{o}-Verblunsky theorem, namely $$\prod_{\alpha \in \mathbb{N}_0^d} (1 - | \gamma_{0,\alpha} |^2) = \exp\left(\int_{\partial\mathbb{B}^d} \log( w(\zeta)) \, {\rm d}\sigma(\zeta)\right).$$ Finally, we identify several classes of weights where one may construct such an $f$ and highlight an explicit example of a weight $w$, residing outside of these classes, where $\prod_{\alpha \in \mathbb{N}_0^d} (1 - |\gamma_{0,\alpha} |^2) \neq \exp\left(\int_{\partial\mathbb{B}^d} \log( w(\zeta)) \, {\rm d}\sigma(\zeta)\right)$.