Abstract. In an upcoming paper, Chang and Skoug used a generalized Brownian motion process to define a generalized analytic Feynman integral and a generalized analytic Fourier-Feynman transform. In this paper we establish several integration by parts formulas involving generalized Feynman integrals, generalized Fourier-Feynman transforms, and the first variation of functionals of the form x , . . . , αn, x ) where α, x denotes the PaleyWiener-Zygmund stochastic integral T 0 α(t)dx(t).

In this paper we first investigate the existence of the generalized Fourier-Feynman transform of the functional F given by F (x) =ν((e 1 , x) ∼ ,. .. , (en, x) ∼), where (e, x) ∼ denotes the Paley-Wiener-Zygmund stochastic integral with x in a very general function space C a,b [0, T ] andν is the Fourier transform of complex measure ν on B(R n) with finite total variation. We then define two sequential transforms. Finally, we establish that the one is to identify the generalized Fourier-Feynman transform and the another transform acts like an inverse generalized Fourier-Feynman transform.

We define two sequential transforms on a function spaceCa,b[0,T]induced by generalized Brownian motion process. We then establish the existence of the sequential transforms for functionals in a Banach algebra of functionals onCa,b[0,T]. We also establish that any one of these transforms acts like an inverse transform of the other transform. Finally, we give some remarks about certain relations between our sequential transforms and other well-known transforms onCa,b[0,T].

In this paper we use a rotation property of Wiener measure to define a very general multiple Fourier-Feynman transform on Wiener space. We then proceed to establish its many algebraic properties as well as to establish several relationships between this generalized multiple transform and the corresponding generalized convolution product.

In this paper we establish a Fubini theorem for functionals on a function space. We then establish some relationships as applications of our Fubini theorem. Finally, we present some historical remarks.