Fourier-Feynman transform, convolution and first variation of functionals on abstract Wiener space

Park, Skoug and Storvick examined various relationships among the first variation, the Fourier-Feynman transform, and the convolution product for functional, on classical Wiener space (C₀[0, 1], m), which belong to some Banach algebra S. In this paper, we extend the above concepts to an abstract Wiener space (B, v) and establish various relationships involving the concepts of Fourier-Feynman transform, convolution, and the first variation for functionals in the Fresnel class F(B) which corresponds to S. Since the Fresnel class F(B) is the abstract Wiener space setting of the Banach algebra S, our results include the above results as special cases.