A Note on General Setting of White Noise Triple and Positive Generalized Functions

[E ]u ⊂ L2(E∗, μ) ⊂ [E ]u, and to characterize white noise test function space [E ]u and generalized function space [E ]u in the series of papers [1],[2],[3],[4],[5]. 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 ([5]). See also [20],[21],[25] for an overview of other recent developments in white noise analysis. This short note is organized as follows. In Section 2, we give a short summary of white noise analysis including AKK’s results. A certain class of convex functions will be introduced to make use of Legendre transformation and dual functions for our purposes. In Section 3, we restate the characterization theorems of the spaces of white noise test and generalized functions given in [3],[5]. In Section 4, we give a quick review of the basic facts on the theory of positive generalized functions [12],[19],[29]. Finally, we discuss the characterization of a Hida measure (Theorem 4.5). In this connection, we present the grey noise and the Poisson noise measures as typical examples inducing positive generalized functions (Examples 4.7 and 4.8, respectively).