• Communications of the Korean Mathematical Society
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• v.19 no.1
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• pp.93-111
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• 2004

In this paper, we use a generalized Brownian motion process to define a generalized Feynman integral and a generalized Fourier-Feynman transform. We also define the concepts of the multiple Lp analytic generalized Fourier-Feynman transform and the generalized convolution product of functional on function space $C_{a,\;b}[0,\;T]$. We then verify the existence of the multiple $L_{p}$ analytic generalized Fourier-Feynman transform for functional on function space that belong to a Banach algebra $S({L_{a,\;b}}^{2}[0, T])$. Finally we establish some relationships between the multiple $L_{p}$ analytic generalized Fourier-Feynman transform and the generalized convolution product for functionals in $S({L_{a,\;b}}^{2}[0, T])$.

• Korean Journal of Mathematics
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• v.6 no.1
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• pp.47-66
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• 1998

In this paper, we introduce an $L_p$ analytic Fourier-Feynman transformation, show the existence of the $L_p$ analytic Fourier-Feynman transforms for a certain class of cylinder functionals on an abstract Wiener space, and investigate its interesting properties. Moreover, we define a convolution product for two functionals on the abstract Wiener space and establish the relationships between the Fourier-Feynman transform for the convolution product of two cylinder functionals and the Fourier-Feynman transform for each functional.

• Bulletin of the Korean Mathematical Society
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• v.35 no.1
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• pp.99-117
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• 1998

Let $(B, H, p_1)$ be an abstract Wiener space and $F(B)$ the Fresnel class on $(B, H, p_1)$ which consists of functionals F of the form : $$ F(x) = \int_{H} exp{i(h,x)^\sim} df(h), x \in B, $$ where $(\cdot, \cdot)^\sim$ is a stochastic inner product between H and B, and f is in $M(H)$, the space of complex Borel measures on H. We introduce an $L_1$ analytic Fourier-Feynman transforms for functionls in $F(B)$. Furthermore, we introduce a convolution on $F(B)$, and then verify the existence of the $L_1$ analytic Fourier-Feynman transform for the convolution product of two functionals in $F(B)$, and we establish the relationships between the $L_1$ analytic Fourier-Feynman tranform of the convolution product for two functionals in $F(B)$ and the $L_1$ analytic Fourier-Feynman transforms for each functional. Finally, we show that most results in [7] follows from our results in Section 3.

• Korean Journal of Mathematics
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• v.9 no.2
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• pp.133-147
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• 2001

In this paper, we introduce the concepts of multiple $L_p$ analytic Fourier-Feynman transform ($1{\leq}p$ < ${\infty})$ and a convolution product of functionals on abstract Wiener space and verify the existence of the multiple $L_p$ analytic Fourier-Feynman transform for functionls in the Fresnel class. Moreover, we verify that the Fresnel class is closed under the $L_p$ analytic Fourier-Feynman transformation and the convolution product, respectively. And we establish some relationships among the multiple $L_p$ analytic Fourier-Feynman transform and the convolution product on the Fresnel class.

• Journal of the Chungcheong Mathematical Society
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• v.19 no.1
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• pp.79-99
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• 2006

In this paper, we define a class of functional defined on a very general function space $C_{a,b}[0,T]$ like a Fresnel class of an abstract Wiener space. We then define the multiple $L_p$ analytic generalized Fourier-Feynman transform and the generalized convolution product of functionals on function space $C_{a,b}[0,T]$. Finally, we establish some relationships between the multiple $L_p$ analytic generalized Fourier-Feynman transform and the generalized convolution product for functionals in $\mathcal{F}(C_{a,b}[0,T])$.

• Journal of the Chungcheong Mathematical Society
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• v.22 no.3
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• pp.481-495
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• 2009

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 $\mathcal{S}$ introduced by Cameron and Storvick. Also Chang, Kim and Yoo extended the above results to an abstract Wiener space for functionals in the Fresnel class $\mathcal{F}(B)$ which corresponds to $\mathcal{S}$. Moreover they introduced the $L_{p}$ analytic Fourier-Feynman transform for functionals on a product abstract Wiener space and then established the above results for functionals in the generalized Fresnel class $\mathcal{F}_{A1,A2}$ containing $\mathcal{F}(B)$. In this paper, we investigate more generalized relationships, between the Fourier-Feynman transform and the convolution product for functionals in $\mathcal{F}_{A1,A2}$, than the above results.

• Korean Journal of Mathematics
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• v.7 no.2
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• pp.183-198
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• 1999

Let $\mathcal{F}(B)$ be the Fresnel class on an abstract Wiener space (B, H, ${\omega}$) which consists of functionals F of the form : $$F(x)={\int}_H\;{\exp}\{i(h,x)^{\sim}\}df(h),\;x{\in}B$$ where $({\cdot}{\cdot})^{\sim}$ is a stochastic inner product between H and B, and $f$ is in $\mathcal{M}(H)$, the space of all complex-valued countably additive Borel measures on H. We introduce the concepts of an $L_p$ analytic Fourier-Feynman transform ($1{\leq}p{\leq}2$) and a convolution product on $\mathcal{F}(B)$ and verify the existence of the $L_p$ analytic Fourier-Feynman transforms for functionls in $\mathcal{F}(B)$. Moreover, we verify that the Fresnel class $\mathcal{F}(B)$ is closed under the $L_p$ analytic Fourier-Feynman transform and the convolution product, respectively. And we investigate some interesting properties for the $n$-repeated $L_p$ analytic Fourier-Feynman transform on $\mathcal{F}(B)$. Finally, we show that several results in [9] come from our results in Section 3.

• Bulletin of the Korean Mathematical Society
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• v.50 no.1
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• pp.217-231
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• 2013

In this paper we establish a Fubini theorem for generalized analytic Feynman integral and $L_1$ generalized analytic Fourier-Feynman transform for the functional of the form $$F(x)=f({\langle}{\alpha}_1,\;x{\rangle},\;{\cdots},\;{\langle}{{\alpha}_m,\;x{\rangle}),$$ where {${\alpha}_1$, ${\cdots}$, ${\alpha}_m$} is an orthonormal set of functions from $L_{a,b}^2[0,T]$. We then obtain several generalized analytic Feynman integration formulas involving generalized analytic Fourier-Feynman transforms.

• Bulletin of the Korean Mathematical Society
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• v.48 no.2
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• pp.223-245
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• 2011

In this paper, we de ne an $L_p$ analytic generalized Fourier Feynman transform and a convolution product of functionals in a Ba-nach algebra $\cal{F}$($C_{a,b}$[0, T]) which is called the Fresnel type class, and in more general class $\cal{F}_{A_1;A_2}$ of functionals de ned on general functio space $C_{a,b}$[0, T] rather than on classical Wiener space. Also we obtain some relationships between the $L_p$ analytic generalized Fourier-Feynman transform and convolution product for functionals in $\cal{F}$($C_{a,b}$[0, T]) and in $\cal{F}_{A_1,A_2}$.

• Honam Mathematical Journal
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• v.35 no.2
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• pp.179-200
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• 2013

Let $C[0,t]$ denote the function space of all real-valued continuous paths on $[0,t]$. Define $X_n:C[0,t]{\rightarrow}\mathbb{R}^{n+1}$ by $Xn(x)=(x(t_0),x(t_1),{\cdots},x(t_n))$, where $0=t_0$ < $t_1$ < ${\cdots}$ < $t_n$ < $t$ is a partition of $[0,t]$. In the present paper, using a simple formula for the conditional expectation given the conditioning function $X_n$, we evaluate the $L_p(1{\leq}p{\leq}{\infty})$-analytic conditional Fourier-Feynman transform and the conditional convolution product of the cylinder functions which have the form $$f((v_1,x),{\cdots},(v_r,x))\;for\;x{\in}C[0,t]$$, where {$v_1,{\cdots},v_r$} is an orthonormal subset of $L_2[0,t]$ and $f{\in}L_p(\mathbb{R}^r)$. We then investigate several relationships between the conditional Fourier-Feynman transform and the conditional convolution product of the cylinder functions.