Products of Wiener Functionals on an Abstract Wiener Space

Abstract

Mikusiński in [1] has proved that the product of the distributions δ (x) and pf. $\frac{1} {{\text{x}}}$ on the one-dimensional Euclidean space ℝ exists in the sense of generalized operations and equals $- \frac{1} {{\text{2}}}\delta \prime \left( {\text{x}} \right)$. This result can be easily extended to the case of an n-dimensional Euclidean space ℝⁿ, i.e. for any $\ell = \left( {\ell _1,\ell _2, \ldots,\ell _{\text{n}} } \right) \in R^{\text{n}},\left( {\ell \ne 0} \right)$,

$$\delta \left( {\left( {\ell,{\text{x}}} \right)} \right) \cdot {\text{pf}}{\text{.}}\frac{1} {{\left( {\ell,{\text{x}}} \right)}} = - \frac{1} {2}\delta \prime \left( {\left( {\ell,{\text{x}}} \right)} \right)\quad {\text{x}} = \left( {{\text{x}}_{\text{1}}, \ldots,{\text{x}}_{\text{n}} } \right) \in R^{\text{n}},$$

where $\left( {\ell,{\text{x}}} \right) = \sum\limits_{{\text{k}} = 1}^{\text{n}} {\ell _{\text{k}} {\text{x}}_{\text{k}}}$.

In this paper we shall try to extend the above results to the case of an infinite dimensional space i.e. an abstract Wiener space.