On Fourier transform of generalized Brownian functionals

Let 4 and 4 be the spaces of generalized Brownian functionals of the white noises B and b, respectively. A Fourier transform from 4 into 4 is defined by [phi](b) = [integral operator]0*: exp[-i [integral operator]1b(t) B(t) dt]: 1), where : : denotes the renormalization with respect to b and [mu] is the standard Gaussian measure on the space 0* of tempered distributions. It is proved that the Fourier transform carries B(t)-differentiation into multiplication by ib(t). The integral representation and the action of[phi] as a generalized Brownian functional are obtained. Some examples of Fourier transform are given.