A note on general setting of white noise triple and positive generalized functions

Abstract

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.