Abstract
We discuss an extension of Toeplitz quantization based on polyanalytic functions. We derive isomorphism theorem for polyanalytic Toeplitz operators between weighted Sobolev-Fock spaces of polyanalytic functions, which are images of modulation spaces under polyanalytic Bargmann transforms. This generalizes well-known results from the analytic setting. Finally, we derive an asymptotic symbol calculus and present an asymptotic expansion of complex Weyl operators in terms of polyanalytic Toeplitz operators.
Highlights
Polyanalytic functions and the associated polyanalytic Bargmann transforms have received a lot of attention in Gabor analysis
This paper is structured as follows: after reviewing some basics about Bargmann transforms and quantization in Sect.2, in Sect. 3 we introduce the idea of true polyanalytic Bargmann transforms as well as polyanalytic Toeplitz quantization Tk(m) of a symbol m : closed subspace F (Cd) → C, where k ∈ Nd indicates the degree of polyanalyticity
Afterwards, we present the polyanalytic generalizations of those spaces and, as a main result, prove an isomorphism theorem for polyanalytic Toeplitz operators
Summary
Polyanalytic functions and the associated polyanalytic Bargmann transforms have received a lot of attention in Gabor analysis.
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