A H\"ormander-Fock space

TL;DR

This paper introduces the H"ormander-Fock space, a new reproducing kernel Hilbert space characterized by a specific moment sequence and entire function, with applications to the bla problem and stochastic processes.

Contribution

It defines and investigates the properties of the H"ormander-Fock space, linking it to polyanalytic functions, kernel functions, and stochastic process theory.

Findings

Characterization of the H"ormander-Fock space via a moment sequence and entire function

Explicit kernel function expression for the space

Application of Bochner-Minlos theorem to a special function in the space

Abstract

In a recent paper we used a basic decomposition property of polyanalytic functions of order $2$ in one complex variable to characterize solutions of the classical $\overline{\partial}$-problem for given analytic and polyanalytic data. Our approach suggested the study of a special reproducing kernel Hilbert space that we call the H\"ormander-Fock space that will be further investigated in this paper. The main properties of this space are encoded in a specific moment sequence denoted by $\eta=(\eta_n)_{n\geq 0}$ leading to a special entire function $\mathsf{E}(z)$ that is used to express the kernel function of the H\"ormander-Fock space. We present also an example of a special function belonging to the class ML introduced recently by Alpay et al. and apply a Bochner-Minlos type theorem to this function, thus motivating further connections with the theory of stochastic processes.