On a stochastic Fourier transformation

Abstract

Given a random function and an orthonormal basis in the , we are concerned with the basic properties of its stochastic Fourier coefficient (SFC) which is defined by the stochastic integral with respect to the Brownian motion, . More precisely we are concerned in this note with the problem of reconstructing the function from its SFCs. We will show as our main result the affirmative answer when the random function belongs to a certain restricted but wide enough subclass, which is the class of causal Wiener functionals.