• 대한수학회보
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• 제60권5호
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• pp.1221-1235
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• 2023

In this paper, we use an infinite dimensional conditioning function to define a conditional Fourier-Feynman transform (CFFT) and a conditional convolution product (CCP) on the Wiener space. We establish the existences of the CFFT and the CCP for bounded functions which form a Banach algebra. We then provide fundamental relationships between the CFFTs and the CCPs.

• 한국수학교육학회지시리즈B:순수및응용수학
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• 제30권2호
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• pp.155-167
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• 2023

In this paper, we use a vector-valued conditioning function to define a conditional Fourier-Feynman transform (CFFT) and a conditional convolution product (CCP) on the Wiener space. We establish the existences of the CFFT and the CCP for bounded functionals which form a Banach algebra. We then provide fundamental relationships between the CFFTs and the CCPs.

• Kyungpook Mathematical Journal
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• 제64권2호
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• pp.219-233
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• 2024

In this paper, we study the generalized Fourier-Feynman transform (GFFT) for functions on the general Wiener space C_a,b[0, T]. We establish an explicit evaluation formula for the analytic GFFT of bounded cylinder functions on C_a,b[0, T]. We start by examining certain cylinder functions which belong in a Banach algebra of bounded functions on C_a,b[0, T]. We then obtain an explicit formula for the analytic GFFT of the bounded cylinder functions.

• 충청수학회지
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• 제24권2호
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• pp.337-344
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• 2011

We provide another unified proof that $A_{\gamma}(G)$ and $A_{\Delta}(G)$ are Banach algebras for a compact group G, where $A_{\gamma}(G)$ and $A_{\Delta}(G)$ are images of the convolution and the twisted convolution, respectively, on $A(G{\times}G)$. Our new approach heavily depends on analysis of co-multiplication on VN(G), the group von-Neumann algebra of G.

• 대한수학회논문집
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• 제22권1호
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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 $F_{A_1,A_2}$.

• 충청수학회지
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• 제22권3호
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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.

• 대한수학회보
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• 제48권2호
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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 $\cal{F}_{A_1,A_2}$.

• 호남수학학술지
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• 제35권1호
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• pp.51-71
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• 2013

In the present paper, we evaluate the analytic conditional Fourier-Feynman transforms and convolution products of unbounded function which is the product of the cylinder function and the function in a Banach algebra which is defined on an analogue o Wiener space and useful in the Feynman integration theories and quantum mechanics. We then investigate the inverse transforms of the function with their relationships and finally prove that th analytic conditional Fourier-Feynman transforms of the conditional convolution products for the functions, can be expressed in terms of the product of the conditional Fourier-Feynman transforms of each function.

• Korean Journal of Mathematics
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• 제23권1호
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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.

• 대한수학회지
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• 제59권2호
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• pp.367-377
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• 2022

Let G be a non-discrete locally compact abelian group, and 𝜇 be a transformable and translation bounded Radon measure on G. In this paper, we construct a Segal algebra S_𝜇(G) in L¹(G) such that the generalized Poisson summation formula for 𝜇 holds for all f ∈ S_𝜇(G), for all x ∈ G. For the definitions of transformable and translation bounded Radon measures and the generalized Poisson summation formula, we refer to L. Argabright and J. Gil de Lamadrid's monograph in 1974.