Seung Jun Chang

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 𝒮(L 2 a, b [0, T]) after it has been multiplied by n linear factors; none of these linear factors belong to 𝒮(L 2 a, b [0, T]). Finally we established some new generalized Feynman integration formulas.

Integration by parts formulas involving generalized Fourier-Feynman transforms on function space

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 F(x) = f( ,..., ) where denotes the Paley-Wiener-Zygmund stochastic integral ∫ T 0 α(t)dx(t).

Convolution products, integral transforms and inverse integral transforms of functionals in L 2(C 0[0, T])

In this paper, we establish several very basic formulas relating convolution products, integral transforms and inverse integral transforms for functionals in L 2(C 0[0, T]). The expansion of functionals in L 2(C 0[0, T]) in terms of Fourier–Hermite functionals plays a key role.

GENERALIZED FOURIER-FEYNMAN TRANSFORM AND SEQUENTIAL TRANSFORMS ON FUNCTION SPACE

In this paper werst investigate the existence of the gener- alized Fourier-Feynman transform of the functional F given by F ( x) = ^ (( e1;x) � ;:::; ( en;x) � ) ; where ( e;x) � denotes the Paley-Wiener-Zyg stochastic integral with x in a very general function space Ca;b(0 ;T ) and ^ is the Fourier transform of complex measure on B( R n ) withnite total variation. We then dene 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.

Parts formulas involving conditional Feynman integrals

In this paper we first obtain a basic formula for the conditional analytic Feynman integral of the first variation of a functional on Wiener space. We then apply this basic result to obtain several integration by parts formulas for conditional analytic Feynman integrals and conditional Fourier-Feynman transforms.

GENERALIZED FOURIER-WIENER FUNCTION SPACE TRANSFORMS

In this paper, we define generalized Fourier-Hermite func- tionals on a function space Ca,b(0, T) to obtain a complete orthonormal set in L2(Ca,b(0, T)) where Ca,b(0, T) is a very general function space. We then proceed to give a necessary and sufficient condition that a func- tional F in L2(Ca,b(0, T)) has a generalized Fourier-Wiener function space transform F√ 2,i (F) also belonging to L2(Ca,b(0, T)).

Relationships Involving Transforms and Convolutions Via the Translation Theorem

In this article, we use the translation theorem to obtain several relationships involving integral transforms and convolution products. In particular, we obtain several useful formulas involving various functionals, which arise naturally in quantum mechanics.

Some basic relationships among transforms, convolution products, first variations and inverse transforms

In this paper we obtain several basic formulas for generalized integral transforms, convolution products, first variations and inverse integral transforms of functionals defined on function space.

A MULTIPLE GENERALIZED FOURIER–FEYNMAN TRANSFORM VIA A ROTATION ON WIENER SPACE

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.

Parts formulas involving conditional generalized Feynman integrals and conditional generalized Fourier–Feynman transforms on function space

In this paper, we first establish a formula for the conditional generalized Feynman integral of the first variation of a functional F(x) with x in a very general function space C a , b [0, T]. We then use this basic formula to obtain several integration by parts formulas for conditional generalized Feynman integrals and conditional generalized Fourier–Feynman transforms. E-mail: jgchoi@dankook.ac.kr E-mail: sejchang@dankook.ac.kr