  Let $\ce^{*}$ be the space of tempered distributions and $\m$ be the standard
Gaussian measure on $\ce^{*}$. Being motivated by the distribution theory on
infinite dimensional space by Cochran, Kuo and Sengupta (CKS) \cite{cks}, Asai,
Kubo and Kuo (AKK) have recently determined the best possible class
$C_{+,{1\over 2},1}^{(2)}$ of functions $u$ to constract white noise triple,
[\ce]_u\subset L^2(\ce^{*},\m) \subset [\ce]^{*}_u, and to characterize white
noise test function space $[\ce]_u$ and generalized function space
$[\ce]_u^{*}$ in the series of papers \cite{akk1}, \citeakk2}, \cite{akk3},
\cite{akk4}, \cite{akk5}. The notion of Legendre transformation plays important
roles to examine relationships between the growth order of holomorphic
functions (S-transform) and the CKS-space of white noise test and generalized
functions. It is well-known that a positive generalized function is induced by
a Hida measure $\nu$ (generalized measure). A Hida measure can be characterized
by integrability conditions on a function inducing the above triple
(\cite{akk5}). See also \cite{kuo99-1}, \cite{kuo99-2}, \cite{ob99} for an
overview of other recent developments in white noise analysis.
