Year: 2024
Type: preprint
Abstract: We establish an abstract Szegő-type limit theorem for Toeplitz operators associated with normalized continuous Parseval frames. Our assumptions consist essentially of a natural normalization of the frame measures and an approximate-identity condition for the normalized frame vectors. Under these hypotheses, the Szegő asymptotics follow from a general argument based on Berezin--Lieb inequalities and convergence of Berezin transforms, independently of the particular geometry of the underlying space. As a principal application, we obtain a Szegő-type limit theorem for Toeplitz operators on weighted Bergman spaces of the unit ball. This setting lies outside the standard Widom-type localization approach because the underlying hyperbolic measure has exponential volume growth. We also obtain Szegő limit theorems for metrizable compact and locally compact Abelian groups, extending the classical compact-group results to more general symbols and recovering recent results for multidimensional tori as special cases. Further applications include Gabor localization operators, Paley--Wiener spaces, eigenvalue distribution, and sampling density results.
Source: arXiv (Cornell University)
Authors: Camper, Trevor, Mitkovski, Mishko
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W4405030201