A note on general setting of white noise triple and positive generalized functions
Abstract
Description
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 $ν$ (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.
Based on the author's talk in RIMS workshop on New development of Infinite dimensional analysis and quantum probability held at RIMS, Kyoto Univ, September 16-17, 1999
Based on the author's talk in RIMS workshop on New development of Infinite dimensional analysis and quantum probability held at RIMS, Kyoto Univ, September 16-17, 1999