The main purpose of this article is to develop the generalized analytic Fourier-Feynman transform theory. We introduce a generalized analytic Fourier-Feynman transform and a multiple generalized analytic Fourier-Feynman transform with respect to Gaussian processes on the function space C a,b [0, T ] induced by a generalized Brownian motion process. We then establish a relationship between these two generalized analytic transforms.

In this paper, we establish a translation theorem for the generalised analytic Feynman integral of functionals that belong to the Banach algebra F (C a,b [0, T ]).2010 Mathematics subject classification: primary 60J65; secondary 28C20, 46G12.

Abstract. In this paper we define a generalized analytic Fourier-Feynman transform associated with Gaussian process on the function space C a,b [0, T ]. We establish the existence of the generalized analytic FourierFeynman transform for certain bounded functionals on C a,b [0, T ]. We then proceed to establish a translation theorem for the generalized transform associated with Gaussian process.

In this article, we introduce a generalized analytic Fourier-Feynman transform and a multiple generalized analytic Fourier-Feynman transform with respect to Gaussian processes on the function space C a,b [0, T] induced by generalized Brownian motion process. We derive a rotation formula for our multiple generalized analytic Fourier-Feynman transform.