Abstract:
Consider the Gelfand triple $mathcal Dsubset L^2(mathbb R_+,dt)subset mathcal D’$, where $mathcal D$ is the space of all smooth functions on $mathbb R_+=[0,infty)$ with compact support and $mathcal D’$ is the dual space of $mathcal D$ with respect to the center space $L^2(mathbb R_+,dt)$. Let $mu$ be a L’evy white noise measure on $mathcal D’$, i.e., a probability measure on $mathcal D’$ such that $X_t=int_0^t omega(t),dt$ is a L’evy process. Thus, $omega(t)=frac d{dt}X_tinmathcal D’$ can be interpreted as a path of a L’evy white noise. We will introduce the notion of a polynomial sequence on $mathcal D’$, and we will single out a class of L’evy white noises for which there exists an orthogonal polynomial sequence on $mathcal D’$. This class includes Gaussian white noise, Poisson point process and gamma random measure. Each system of such orthogonal polynomials can be interpreted as a Sheffer sequence on $mathcal D’$. The classical umbral calculus studies Sheffer polynomial sequences in the one-dimensional or multivariate setting. Extending Grabiner’s result related to the one-dimensional umbral calculus, we will construct a class of spaces of entire functions on $mathcal D’_{mathbb C}$—the complexification of $mathcal D$—spanned by Sheffer polynomial sequences on $mathcal D’$. This will, in particular, extend the well-known characterization of the Hida test space of Gaussian white noise as a space of entire functions on $mathcal D_{mathbb C}’$.