An Analytic Characterization of p , q -White Noise Functionals

Abstract

In this paper, a characterization theorem for the $\mathcal{S}$ -transform of infinite dimensional distributions of noncommutative white noise corresponding to the $\left(p,q\right)$ -deformed quantum oscillator algebra is investigated. We derive a unitary operator $U$ between the noncommutative $L^2$ -space and the $\left(p,q\right)$ -Fock space which serves to give the construction of a white noise Gel’fand triple. Next, a general characterization theorem is proven for the space of $\left(p,q\right)$ -Gaussian white noise distributions in terms of new spaces of $\left(p,q\right)$ -entire functions with certain growth rates determined by Young functions and a suitable $\left(p,q\right)$ -exponential map.