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

Abstract: Given a measure $μ$ on the unit sphere $\partial\mathbb{B}^d$ in $\mathbb{C}^d$ with Lebesgue decomposition ${\rm d} μ= w \, {\rm d} σ+ {\rm d} μs$, with respect to the rotation-invariant Lebesgue measure $σ$ on $\partial \mathbb{B}^d$, we introduce notions of orthogonal polynomials $(\varphiα){α\in \mathbb{N}_0^d}$, Verblunsky coefficients $(γ{α,β}){α,β\in \mathbb{N}_0^d}$, and an associated Christoffel function $λ{\infty}^{(d)}(z; {\rm d} μ)$, 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 ${\rm supp}\, μs$ is discrete, then the aforementioned quantities depend only on the absolutely continuous part of $μ$. If, in addition to ${\rm supp}\, μ_s$ being discrete, one is able to find $f \in H^{\infty}(\mathbb{B}^d)$ such that $f(0) = 1$ and $$\int{\partial \mathbb{B}^d} |f(ζ)|^2 w(ζ) {\rm d}σ(ζ) \leq \exp\left( \int_{\partial \mathbb{B}^d} \log(w(ζ)) \, {\rm d}σ(ζ) \right),$$ then we establish a $d$-variate Szegő-Verblunsky theorem, namely $$\prod_{α\in \mathbb{N}0^d} (1 - | γ{0,α} |^2) = \exp\left(\int_{\partial\mathbb{B}^d} \log( w(ζ)) \, {\rm d}σ(ζ)\right).$$ 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 $\prod_{α\in \mathbb{N}0^d} (1 - |γ{0,α} |^2) \neq \exp\left(\int_{\partial\mathbb{B}^d} \log( w(ζ)) \, {\rm d}σ(ζ)\right)$.