• Communications of the Korean Mathematical Society
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• v.22 no.1
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• pp.75-90
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• 2007

Huffman, Park and Skoug introduced various results for the $L_p$ analytic Fourier-Feynman transform and the convolution for functionals on classical Wiener space which belong to some Banach algebra S introduced by Cameron and Strovic. Also Chang, Kim and Yoo extended the above results to an abstract Wiener space for functionals in the Fresnel class F(B) which corresponds to S. Recently Kim, Song and Yoo investigated more generalized relationships between the Fourier-Feynman transform and the convolution product for functionals in a generalized Fresnel class $F_{A_1,A'_2}$ containing F(B). In this paper, we establish various interesting relationships and expressions involving the first variation and one or two of the concepts of the Fourier-Feynman transform and the convolution product for functionals in $F_{A_1,A_2}$.

• Journal of the Korean Mathematical Society
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• v.53 no.3
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• pp.709-723
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• 2016

Let C[0, t] denote a generalized Wiener space, the space of real-valued continuous functions on the interval [0, t] and define a random vector $Z_n:C[0,t]{\rightarrow}{\mathbb{R}}^n$ by $Zn(x)=(\int_{0}^{t_1}h(s)dx(s),{\cdots},\int_{0}^{t_n}h(s)dx(s))$, where 0 < $t_1$ < ${\cdots}$ < $t_n$ < t is a partition of [0, t] and $h{\in}L_2[0,t]$ with $h{\neq}0$ a.e. In this paper we will introduce a simple formula for a generalized conditional Wiener integral on C[0, t] with the conditioning function $Z_n$ and then evaluate the generalized analytic conditional Wiener and Feynman integrals of the cylinder function $F(x)=f(\int_{0}^{t}e(s)dx(s))$ for $x{\in}C[0,t]$, where $f{\in}L_p(\mathbb{R})(1{\leq}p{\leq}{\infty})$ and e is a unit element in $L_2[0,t]$. Finally we express the generalized analytic conditional Feynman integral of F as two kinds of limits of non-conditional generalized Wiener integrals of polygonal functions and of cylinder functions using a change of scale transformation for which a normal density is the kernel. The choice of a complete orthonormal subset of $L_2[0,t]$ used in the transformation is independent of e and the conditioning function $Z_n$ does not contain the present positions of the generalized Wiener paths.

• Korean Journal of Mathematics
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• v.23 no.1
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• pp.47-64
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• 2015

We express generalized Fourier-Feynman transform and convolution product of functionals in a Banach algebra $\mathcal{S}(L^2_{a,b}[0,T])$ as limits of function space integrals on $C_{a,b}[0,T]$. Moreover we obtain change of scale formulas for function space integrals related with generalized Fourier-Feynman transform and convolution product of these functionals.

• Korean Journal of Mathematics
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• v.26 no.3
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• pp.387-403
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• 2018

Shifting, scaling and modulation proprerties for the convolution product of the Fourier-Feynman transform of functionals in a generalized Fresnel class ${\mathcal{F}}_{A1,A2}$ are given. These properties help us to obtain convolution product of new functionals from the convolution product of old functionals which we know their convolution product.

• Communications of the Korean Mathematical Society
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• v.36 no.4
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• pp.685-704
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• 2021

The purpose of this paper is to establish a very general Cameron-Storvick theorem involving the generalized analytic Feynman integral of functionals on the product function space C²_a,b[0, T]. The function space C_a,b[0, T] can be induced by the generalized Brownian motion process associated with continuous functions a and b. To do this we first introduce the class ${\mathcal{F}}^{a,b}_{A_1,A_2}$ of functionals on C²_a,b[0, T] which is a generalization of the Kallianpur and Bromley Fresnel class ${\mathcal{F}}_{A_1,A_2}$. We then proceed to establish a Cameron-Storvick theorem on the product function space C²_a,b[0, T]. Finally we use our Cameron-Storvick theorem to obtain several meaningful results and examples.

• The Pure and Applied Mathematics
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• v.2 no.2
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• pp.103-110
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• 1995

In their recent paper[2], they show that the existence theory for the analytic operator-valued Feynman path integral can be extended by making use of recent developments in the theory of Dirichlet forms and Markov process. In this field, there is the necessity of studying certain generalized functionals of the process (of Feynman-Kac type). Their study have been concerned with Feynman-Kac type functionals related with smooth measures associated with the classical Dirichlet form (associated with the Laplacian).(omitted)

• Journal of the Korean Mathematical Society
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• v.50 no.5
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• pp.1105-1127
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• 2013

Let C[0, $t$] denote the function space of real-valued continuous paths on [0, $t$]. Define $X_n\;:\;C[0,t]{\rightarrow}\mathbb{R}^{n+1}$ and $X_{n+1}\;:\;C[0,t]{\rightarrow}\mathbb{R}^{n+2}$ by $X_n(x)=(x(t_0),x(t_1),{\ldots},x(t_n))$ and $X_{n+1}(x)=(x(t_0),x(t_1),{\ldots},x(t_n),x(t_{n+1}))$, respectively, where $0=t_0 <; t_1 <{\ldots} < t_n < t_{n+1}=t$. In the present paper, using simple formulas for the conditional expectations with the conditioning functions $X_n$ and $X_{n+1}$, we evaluate the $L_p(1{\leq}p{\leq}{\infty})$-analytic conditional Fourier-Feynman transforms and the conditional convolution products of the functions, which have the form $fr((v_1,x),{\ldots},(v_r,x)){\int}_{L_2}_{[0,t]}\exp\{i(v,x)\}d{\sigma}(v)$ for $x{\in}C[0,t]$, where $\{v_1,{\ldots},v_r\}$ is an orthonormal subset of $L_2[0,t]$, $f_r{\in}L_p(\mathbb{R}^r)$, and ${\sigma}$ is the complex Borel measure of bounded variation on $L_2[0,t]$. We then investigate the inverse conditional Fourier-Feynman transforms of the function and prove that the analytic conditional Fourier-Feynman transforms of the conditional convolution products for the functions can be expressed by the products of the analytic conditional Fourier-Feynman transform of each function.

• Journal of the Chungcheong Mathematical Society
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• v.24 no.3
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• pp.517-528
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• 2011

Cameron and Storvick introduced change of scale formulas for Wiener integrals of bounded functions in the Banach algebra $\mathcal{S}$ of analytic Feynman integrable functions on classical Wiener space. Yoo and Skoug extended this result to an abstract Wiener space. Also Yoo, Song, Kim and Chang established a change of scale formula for Wiener integrals of functions on abstract Wiener space which need not be bounded or continuous. In this paper, we investigate a change of scale formula for generalized Wiener integrals of various functions on classical Wiener space.

• Journal of the Korean Mathematical Society
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• v.55 no.1
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• pp.147-160
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• 2018

In this article, we establish translation theorems for the analytic Fourier-Feynman transform of functionals in non-stationary Gaussian processes on Wiener space. We then proceed to show that these general translation theorems can be applied to two well-known classes of functionals; namely, the Banach algebra S introduced by Cameron and Storvick, and the space ${\mathcal{B}}^{(P)}_{\mathcal{A}}$ consisting of functionals of the form $F(x)=f({\langle}{\alpha}_1,x{\rangle},{\ldots},{\langle}{\alpha}_n,x{\rangle})$, where ${\langle}{\alpha},x{\rangle}$ denotes the Paley-Wiener-Zygmund stochastic integral ${\int_{0}^{T}}{\alpha}(t)dx(t)$.

• Communications of the Korean Mathematical Society
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• v.21 no.1
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• pp.117-133
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• 2006

Cameron and Storvick discovered change of scale formulas for Wiener integrals of bounded functions in a Banach algebra S of analytic Feynman integrable functions on classical Wiener space. Yoo and Skoug extended these results to abstract Wiener space for a generalized Fresnel class $F_{A1,A2}$ containing the Fresnel class F(B) which corresponds to the Banach algebra S on classical Wiener space. In this paper, we present a change of scale formula for Wiener integrals of various functions on $B^2$ which need not be bounded or continuous.