Matrix Wiener transform

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] and [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 L2(Rn,dμc), where f   belongs to the class of complex valued polynomials on RnRn, and dμc being the Gaussian measure on RnRn as a unitary map [5].