Characterization of Hida Measures in White Noise Analysis

In the recent paper [3] by Asai et al., the growth order of holomorphic functions on a nuclear space has been considered. For this purpose, certain classes of growth functions u are introduced and many properties of Legendre transform of such functions are investigated. In [4], applying Legendre transform of u under the conditions (U0), (U2) and (U3) (see §2), the Gel’fand triple [E ]u ⊂ (L) ⊂ [E ]u associated with a growth function u is constructed. The main purpose of this work is to prove Theorem 4.4, so-called, the characterization theorem of Hida measures (generalized measures). As examples of such measures, we shall present the Poisson noise measure and the Grey noise measure in Example 4.5 and 4.6, respectively. The present paper is organized as follows. In §2, we give a quick review of some fundamental results in white noise analysis and introduce the notion of Legendre transform utilized by Asai et al. in [3],[4]. In §3, we simply cite some useful properties of the Legendre transform from [3]. In §4, we discuss the characterization of Hida measures (generalized measures).