A biorthogonal approach to the infinite dimensional fractional Poisson measure

Abstract

In this paper we use a biorthogonal approach to the analysis of the infinite dimensional fractional Poisson measure $π_σ^β$, $0<β\leq1$, on the dual of Schwartz test function space $\mathcal{D}'$. The Hilbert space $L^{2}(π_σ^β)$ of complex-valued functions is described in terms of a system of generalized Appell polynomials $\mathbb{P}^{σ,β,α}$ associated to the measure $π_σ^β$. The kernels $C_{n}^{σ,β}(\cdot)$, $n\in\mathbb{N}_{0}$, 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 $\mathbb{P}^{σ,β,α}$, there is a generalized dual Appell system $\mathbb{Q}^{σ,β,α}$ that is biorthogonal to $\mathbb{P}^{σ,β,α}$. The test and generalized function spaces associated to the measure $π_σ^β$ are completely characterized using an integral transform as entire functions.