2025/12/11 by Connor J. Gauntlett, Gauntlett, Connor J., David P. Kimsey +1
Mathematics · #32A10 #32E30 #42C05 #Complex Variables (math.CV) #FOS: Mathematics #Functional Analysis (math.FA) #Holomorphic and Operator Theory #Mathematical functions and polynomials #Random Matrices and Applications
paper · pdf · doi:10.48550/arxiv.2512.10528
openalex publication_date 2025/12/11 · openalex created_date 2025/12/13 · openalex updated_date 2026/07/28
Given a measure μ on the unit sphere ∂\mathbbBd in ℂd with Lebesgue decomposition \rm d μ= w \rm d σ+ \rm d μs, with respect to the rotation-invariant Lebesgue measure σ on ∂ \mathbbBd, we introduce notions of orthogonal polynomials (φα)α∈ ℕ0d, Verblunsky coefficients (γα,β)α,β∈ ℕ0d, and an associated Christoffel function λ∞(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 μs being discrete, one is able to find f ∈ H∞(\mathbbBd) such that f(0) = 1 and ∫_∂ \mathbbBd |f(ζ)|2 w(ζ) \rm dσ(ζ) ≤ exp( ∫_∂ \mathbbBd log(w(ζ)) \rm dσ(ζ) ), then we establish a d-variate Szegő-Verblunsky theorem, namely ∏α∈ ℕ0d (1 - | γ0,α |2) = exp(∫_∂\mathbbBd log( w(ζ)) \rm dσ(ζ)). 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 ∏α∈ ℕ0d (1 - |γ0,α |2) ≠ exp(∫_∂\mathbbBd log( w(ζ)) \rm dσ(ζ)).