Wick analysis for Bernoulli noise functionals (Q2443812)

Summary: A Gel'fand triple \(S(\Omega) \subset L^2(\Omega) \subset S^\ast(\Omega)\) is constructed of functionals of \(Z\), where \(Z=(Z_n)_{n\in \mathbb N}\) is an appropriate Bernoulli noise on a probability space \((\Omega, \mathcal F, \mathbb P)\). Characterizations are given to both \(S(\Omega)\) and \(S^\ast(\Omega)\). It is also shown that a Wick-type product can be defined on \(S^\ast(\Omega)\) and, moreover, \(S^\ast(\Omega)\) forms a commutative algebra with the product. Finally, a transform called \(S\)-transform is defined on \(S^\ast(\Omega)\) and its continuity as well as other properties are examined.