Conditional integral transforms and conditional convolution products on a function space

Abstract

Let C[0, t] denote the function space of all real-valued continuous paths on [0, t]. Define and by X n (x)=(x(t 0), x(t 1), …, x(t n )) and X n+1(x)=(x(t 0), x(t 1), …, x(t n ), x(t n+1)), where 0=t 0<t 1<⋯<t n <t n+1=t. In this paper, using two simple formulas for the conditional expectations with the conditioning functions X n and X n+1, we evaluate the conditional analytic Fourier–Feynman transforms and the conditional convolution products of the cylinder functions which have the form f((v 1, x), …, (v r , x)) for x∈C[0, t], where {v 1, …, v r } is an orthonormal subset of L 2[0, t] and . We finally show that the conditional analytic Fourier–Feynman transforms of the conditional convolution products for the cylinder functions can be expressed in terms of the products of the conditional analytic Fourier–Feynman transform of each function.