Generalized Fourier-Feynman transforms and a first variation on function space

In this paper we use a generalized Brownian motion process to define a generalized Feynman integral and a generalized Fourier-Feynman transform. We establish a translation theorem and use it to express the generalized Feynman integral of the first variation of a functional F in terms of the generalized Feynman integral of F multiplied by a linear factor. We establish some integration by parts formulas for generalized Feynman integrals and transforms. We also find the generalized Fourier-Feynman transform of a functional F belonging to a Banach algebra (L-a,b(2) [0, T ]) after it has been multiplied by n linear factors; none of these linear factors belong to (L-a,b(2) [0, T ]). Finally we established some new generalized Feynman integration formulas.