General characterization theorems and intrinsic topologies in white noise analysis

Abstract

Let $u$ be a positive continuous function on $[0,\infty )$ satisfying the conditions: (i) ${\lim }_{r\to \infty }{r}^{-1/2}\log u(r)=\infty$, (ii) ${\inf }_{r\ge 0}u(r)=1$, (iii) $\lim_{r\to \infty}\break r^{-1}\log u(r)<\infty$, (iv) the function $\log u(x^2),x\ge 0$, is convex. A Gel'fand triple $[\ce]_{u} \subset (L^{2}) \subset [\ce]_{u}^{*}$ is constructed by making use of the Legendre transform of $u$ discussed in \cite {akk3}. We prove a characterization theorem for generalized functions in $[\ce]_{u}^{*}$ and also for test functions in $[\ce]_{u}$ in terms of their $S$-transforms under the same assumptions on $u$. Moreover, we give an intrinsic topology for the space$[\ce]_{u}$ of test functions and prove a characterization theorem for measures. We briefly mention the relationship between our method and a recent work by Gannoun et al.\cite{ghor}. Finally, conditions for carrying out white noise operator theory and Wick products are given.

To appear in Hiroshima Math. J. 31, Louisiana state university preprint (2000)