  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) $\inf_{r\geq
0} u(r)=1$, (iii) $\lim_{r\to \infty}\break r^{-1}\log u(r)<\infty$, (iv) the
function $\log u(x^{2}), x\geq 0$, 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.
