Reproducing kernel Hilbert spaces of polyanalytic functions of infinite order

TL;DR

This paper introduces and analyzes reproducing kernel Hilbert spaces of polyanalytic functions of infinite order, exploring their kernels, transforms, and operators, extending finite-order polyanalytic space theory.

Contribution

It develops the theory of infinite-order polyanalytic reproducing kernel Hilbert spaces, including kernel functions, transforms, and shift operators, expanding the mathematical framework beyond finite-order cases.

Findings

Kernel function for the infinite-order polyanalytic space is $e^{zar{w}+ar{z}w}$.

Connections established between these spaces and finite-order polyanalytic Fock spaces.

Backward shift operators are defined and studied within these spaces.

Abstract

In this paper we introduce reproducing kernel Hilbert spaces of polyanalytic functions of infinite order. First we study in details the counterpart of the Fock space and related results in this framework. In this case the kernel function is given by $\displaystyle e^{z\overline{w}+\overline{z}w}$ which can be connected to kernels of polyanalytic Fock spaces of finite order. Segal-Bargmann and Berezin type transforms are also considered in this setting. Then, we study the reproducing kernel Hilbert spaces of complex-valued functions with reproducing kernel $\displaystyle\frac{1}{(1-z\overline{w})(1-\overline{z}w)}$ and $\displaystyle\frac{1}{1-2{\rm Re}\, z\overline{w}}$. The corresponding backward shift operators are introduced and investigated.