Matrix Wiener transform

Abstract

In this paper we analyse some properties of the matricial expression of the Fourier–Wiener transform, a matrix transform firstly treated by Cameron and Martin for analytic functions [3] , [4] . Here the referred properties are a composition formula, a Parseval relation and an inversion formula, which, according to Segal (1956) [13] extends an unitary explicit integral representation of the second quantization for one integral operator of the Wiener transform [12] . This work includes the unitary extension of the transform to L 2 ( R n , d μ c ) , where f belongs to the class of complex valued polynomials on R n , and dμ c being the Gaussian measure on R n as a unitary map [5] .