A biorthogonal approach to the infinite dimensional fractional Poisson measure

TL;DR

This paper develops a biorthogonal framework for analyzing the infinite dimensional fractional Poisson measure, describing its function spaces and polynomial systems using advanced mathematical tools like Appell polynomials and Stirling operators.

Contribution

It introduces a novel biorthogonal approach and characterizes the associated function spaces for the fractional Poisson measure in infinite dimensions.

Findings

Characterization of the Hilbert space $L^{2}( ext{measure})$ using generalized Appell polynomials.

Expression of kernels in terms of Stirling operators and falling factorials.

Complete description of test and generalized function spaces via integral transform.

Abstract

In this paper we use a biorthogonal approach to the analysis of the infinite dimensional fractional Poisson measure $\pi_{\sigma}^{\beta}$, $0<\beta\leq1$, on the dual of Schwartz test function space $\mathcal{D}'$. The Hilbert space $L^{2}(\pi_{\sigma}^{\beta})$ of complex-valued functions is described in terms of a system of generalized Appell polynomials $\mathbb{P}^{\sigma,\beta,\alpha}$ associated to the measure $\pi_{\sigma}^{\beta}$. The kernels $C_{n}^{\sigma,\beta}(\cdot)$, $n\in\mathbb{N}_{0}$, of the monomials may be expressed in terms of the Stirling operators of the first and second kind as well as the falling factorials in infinite dimensions. Associated to the system $\mathbb{P}^{\sigma,\beta,\alpha}$, there is a generalized dual Appell system $\mathbb{Q}^{\sigma,\beta,\alpha}$ that is biorthogonal to $\mathbb{P}^{\sigma,\beta,\alpha}$. The test and generalized function spaces associated to the measure $\pi_{\sigma}^{\beta}$ are completely characterized using an integral transform as entire functions.