Salvato in:
| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| Soggetti: | |
| Accesso online: | https://arxiv.org/abs/2512.10528 |
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Sommario:
- Given a measure $μ$ on the unit sphere $\partial\mathbb{B}^d$ in $\mathbb{C}^d$ with Lebesgue decomposition ${\rm d} μ= w \, {\rm d} σ+ {\rm d} μ_s$, with respect to the rotation-invariant Lebesgue measure $σ$ on $\partial \mathbb{B}^d$, we introduce notions of orthogonal polynomials $(φ_α)_{α\in \mathbb{N}_0^d}$, 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 ${\rm supp}\, μ_s$ 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 $\prod_{α\in \mathbb{N}_0^d} (1 - |γ_{0,α} |^2) \neq \exp\left(\int_{\partial\mathbb{B}^d} \log( w(ζ)) \, {\rm d}σ(ζ)\right)$.