Hyun Soo Chung

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.

FOURIER-TYPE FUNCTIONALS ON WIENER SPACE

In this paper we dene the Fourier-type functionals via the Fourier transform on Wiener space. We investigate some properties of the Fourier-type functionals. Finally, we establish integral transform of the Fourier-type functionals which also can be expressed by other Fourier- type functionals.

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.

Some Applications of the Spectral Theory for the Integral Transform Involving the Spectral Representation

In many previous papers, an integral transform was just considered as a transform on appropriate function spaces. In this paper we deal with the integral transform as an operator on a function space. We then apply various operator theories to . Finally we give an application for the spectral representation of a self-adjoint operator which plays a key role in quantum mechanics.

GENERALIZED ANALYTIC FEYNMAN INTEGRAL VIA FUNCTION SPACE INTEGRAL OF BOUNDED CYLINDER FUNCTIONALS

In this paper, we use a generalized Brownian motion to dene a generalized analytic Feynman integral. We then obtain some results for the generalized analytic Feynman integral of bounded cylinder functionals of the form F(x) = ^ ((g1;x) � ;:::;(gn;x) � ) dened on a very general function space Ca;b(0;T). We also present a change of scale formula for function space integrals of such cylinder func- tionals.

AN APPROACH TO SOLUTION OF THE SCHR¨ ODINGER EQUATION USING FOURIER-TYPE FUNCTIONALS

Abstract. In this paper, we consider the Fourier-type functionals onWiener space. We then establish the analytic Feynman integrals involv-ing the ⋄-convolutions. Further, we give an approach to solution of theSchro¨dinger equation via Fourier-type functionals. Finally, we use this ap-proach to obtain solutions of the Schro¨dinger equations for harmonic oscil-lator and double-well potential. The Schro¨dinger equations for harmonicoscillator and double-well potential are meaningful subjects in quantummechanics. 1. IntroductionLet C 0 [0,T] denote the one-parameter Wiener space, that is, the space ofcontinuous real-valued functions xon [0,T] with x(0) = 0. In 1948, Feynmanassumed the existence of an integral over a space of paths and used this integralin a formal way in his approach to quantum mechanics [9]. A number ofmathematicians have attempted to give rigorously meaningful definitions ofthe Feynman integral with appropriate existence theorems and have expressedsolutions of the Schr¨odinger equation in terms of their integrals. One of theseapproaches is based on the similarity between the Wiener and the Feynmanintegrals, where procedures are developed to obtain the Feynman integralsfrom the Wiener integrals by an analytic extension from the real axis to theimaginary axis.Consider a differential equation(1.1)∂∂tψ(u,t) =12λ∆ψ(u,t) −V(u)ψ(u,t)with the initial condition ψ(u,0) = ϕ(u), where ∆ is the Laplacian and V is anappropriate potential function. For λ>0, this is the diffusion equation with

Sequential Generalized Transforms on Function Space

We define two sequential transforms on a function space induced by generalized Brownian motion process. We then establish the existence of the sequential transforms for functionals in a Banach algebra of functionals on . 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 on .

CONDITIONAL TRANSFORM WITH RESPECT TO THE GAUSSIAN PROCESS INVOLVING THE CONDITIONAL CONVOLUTION PRODUCT AND THE FIRST VARIATION

In this paper, we define a conditional transform with respect to the Gaussian process, the conditional convolution product and the first variation of functionals via the Gaussian process. We then examine vari- ous relationships of the conditional transform with respect to the Gauss- ian process, the conditional convolution product and the first variation for functionals F in S� (5, 8).