Analytic version of test functionals, Fourier transform, and a characterization of measures in white noise calculus

Abstract

It is shown that the space ( J ) of test white noise functionals has an analytic version A ∞ which is an algebra as well as a topological linear space topologized by the projective limit of a sequence { A p :pϵ N } of Banach spaces with respective norm given by ∥ƒ∥ A p =sup Zϵ CJ − p {|ƒ( z)|exp[−2 −1∥ z∥] p 2]}, where SJ −p denotes the complexification of J −p . Furthermore, it is shown that the A ∞-topology and the ( J )-topology are equivalent. In the course of the proof, it is also shown that the space ( J p ) is isometrically isomorphic to the Bargmann-Segal analytic functions on bJ −p under S-transform. Employing this new version, we are able to define the Fourier transform of a generalized white noise functional as the adjoint of Fourier-Wiener transform T 1, −i with parameter (1, − i), and, moreover, we show that every measure in ( J ) ∗ always satisfies a certain “growth condition.”