Generalized integral transforms with the translation operator on function space

Main goal of this paper is to establish various basic formulas for the generalized integral transform involving the generalized convolution product. In order to establish these formulas, we use the translation operator which was introduced in [9]. It was not easy to establish basic formulas for the generalized integral transforms because the generalized Brownian motion process used in this paper has the nonzero mean function. In this paper, we can easily establish various basic formulas for the generalized integral transform involving the generalized convolution product via the translation operator.

Generalized analytic Fourier-Feynman transforms with respect to Gaussian processes on function space

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 Ca,b[0,T] induced by generalized Brownian motion process. We derive a rotation formula for our multiple generalized analytic Fourier-Feynman transform.

Sequential Generalized Transforms on Function Space

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].

Relationships of convolution products, generalized transforms, and the first variation on function space

We use a generalized Brownian motion process to define the generalized Fourier-Feynman transform, the convolution product, and the first variation. We then examine the various relationships that exist among the first variation, the generalized Fourier-Feynman transform, and the convolution product for functionals on function space that belong to a Banach algebraS(Lab[0,T]). These results subsume similar known results obtained by Park, Skoug, and Storvick (1998) for the standard Wiener process.

Conditional generalized analytic Feynman integrals and a generalized integral equation

We use a generalized Brownian motion process to define a generalized Feynman integral and a conditional generalized Feynman integral. We then establish the existence of these integrals for various functionals. Finally we use the conditional generalized Feynman integral to derive a Schrödinger integral equation.