Il Yong Lee

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).

Relationships between the ∗w-product and the L1 analytic Fourier–Feynman transforms

ABSTRACT In this paper, we introduce the concept of the -product which is a very useful tool to obtain the analytic Fourier–Feynman transform. We then use this concept to obtain various integration formulas involving the -product, the convolution product and the first variation.

Generalized conditional transform with respect to the Gaussian process on function space

In this paper, we define the generalized conditional transform with respect to the Gaussian process. We then establish the integration formulas for the generalized conditional transform with respect to the Gaussian process in terms of the generalized conditional ⋄-product and the first variation. Also, we show that the generalized conditional transform with respect to the Gaussian process for the first variation of F can be expressed in terms of the ordinary function space integral of F multiplied by a linear factor.

A FUBINI THEOREM FOR GENERALIZED ANALYTIC FEYNMAN INTEGRAL ON FUNCTION SPACE

In this paper we establish a Fubini theorem for generalized analytic Feynman integral and L1 generalized analytic Fourier-Feynman transform for the functional of the form F(x) = f(h�1,xi,...,hm,xi), where f�1,...,�mg is an orthonormal set of functions from L2 (0,T). We then obtain several generalized analytic Feynman integration formulas involving generalized analytic Fourier-Feynman transforms.

GENERALIZED ANALYTIC FOURIER-FEYNMAN TRANSFORMS AND CONVOLUTIONS ON A FRESNEL TYPE CLASS

In this paper, we dene an Lp analytic generalized Fourier- Feynman transform and a convolution product of functionals in a Ba- nach algebra F ( Ca;b(0 ;T )) which is called the Fresnel type class, and in more general class FA1;A2 of functionals dened on general function space Ca;b(0 ;T ) rather than on classical Wiener space. Also we obtain some relationships between the Lp analytic generalized Fourier-Feynman transform and convolution product for functionals in F ( Ca;b(0 ;T )) and in FA1;A2 .

A new approach method to obtain the L1 generalized analytic Fourier–Feynman transform

ABSTRACT In this paper, we first introduce the concept of the ⊗-product on function space. We then proceed to use this concept to obtain several integration formulas. In addition, we establish various relationships which exist. Also, we establish the relationships among the ⊗-product, the generalized convolution product and the first variation.

Series expansions of the transform with respect to the Gaussian process

In this paper, we consider the Fourier-type functionals introduced in [Chung HS, Tuan VK. Fourier-type functionals on Wiener space. Bull Korean Math Soc. 2012;49:609–619] and used in [Lee IY, Chung HS, Chang SJ. Series expansions of the analytic Feynman integral for the Fourier-type functional. J Korean Soc Math Educ Ser B. 2012;19:87–102]. We first establish the existence of the transform with respect to the Gaussian process for the Fourier-type functionals. We then obtain the various relationships for the transform with respect to the Gaussian process of the Fourier-type functionals which involved the ⋄-product and in addition, establish various integration formulas.