Analytic Fourier--Feynman transforms and convolution type operations associated with Gaussian processes on Wiener space

In this paper we introduce the concept of a convolution type operation of functionals on Wiener space. It contains several kinds of the concepts of convolution products on Wiener space, which have been studied by many authors. We then investigate fundamental relationships between generalized analytic Fourier--Feynman transforms and convolution type operations. Both of the generalized analytic Fourier--Feynman transform of the convolution type operation and the convolution type operation of the generalized analytic Fourier--Feynman transforms are represented as a product of the generalized analytic Fourier--Feynman transforms.