• Bulletin of the Korean Mathematical Society
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• v.41 no.1
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• pp.73-93
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• 2004

In [10], Chang and Skoug used a generalized Brownian motion process to define a generalized analytic Feynman integral and a generalized analytic Fourier-Feynman transform. In this paper we define the conditional generalized Fourier-Feynman transform and conditional generalized convolution product on function space. We then establish some relationships between the conditional generalized Fourier-Feynman transform and conditional generalized convolution product for functionals on function space that belonging to a Banach algebra.

• 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 $S({L_{a,\;b}}^{2}[0, T])$.

• Journal of the Chungcheong Mathematical Society
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• v.21 no.2
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• pp.231-246
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• 2008

In this paper, we use a generalized Brownian motion process to define a generalized analytic Feynman integral. We then establish several integration formulas for generalized analytic Feynman integrals generalized analytic Fourier-Feynman transforms and generalized integral transforms of functionals in the class of functionals ${\mathbb{E}}_0$. Finally, we use these integration formulas to obtain several generalized Feynman integrals involving the generalized analytic Fourier-Feynman transform and the generalized integral transform of functionals in ${\mathbb{E}}_0$.

• Communications of the Korean Mathematical Society
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• v.26 no.2
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• pp.273-289
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• 2011

In this paper we dene the concept of a conditional generalized Fourier-Feynman transform on very general function space $C_{a,b}$[0, T]. We then establish the existence of the conditional generalized Fourier-Feynman transform for functionals in a Fresnel type class. We also obtain several results involving the conditional transform. Finally we present functionals to apply our results. The functionals arise naturally in Feynman integration theories and quantum mechanics.

• Korean Journal of Mathematics
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• v.29 no.2
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• pp.321-332
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• 2021

This paper is a further development of the recent results by the authors on the generalized sequential Fourier-Feynman transform for functionals in a Banach algebra Ŝ and some related functionals. We investigate various relationships between the generalized sequential Fourier-Feynman transform and the generalized sequential convolution product of functionals. Parseval's relation for the generalized sequential Fourier-Feynman transform is also given.

• Korean Journal of Mathematics
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• v.25 no.3
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• pp.335-347
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• 2017

Time shifting and frequency shifting proprerties for the Fourier-Feynman transform of functionals in a generalized Fresnel class ${\mathcal{F}}_{A_1,A_2}$ are given. We discuss scaling and modulation proprerties for the Fourier-Feynman transform. These properties help us to obtain Fourier-Feynman transforms of new functionals from the Fourier-Feynman transforms of old functionals which we know their Fourier-Feynman transforms.

• Korean Journal of Mathematics
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• v.31 no.4
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• pp.521-536
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• 2023

This paper is a further development of the recent results by the author and coworker on the generalized sequential Fourier-Feynman transform for functionals in a Banach algebra Ŝ and some related functionals. We establish existence of the generalized first variation of these functionals. Also we investigate various relationships between the generalized sequential Fourier-Feynman transform, the generalized sequential convolution product and the generalized first variation of the functionals.

• The Pure and Applied Mathematics
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• v.28 no.4
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• pp.281-296
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• 2021

In this paper, we suggest a representation for an inverse transform of the generalized Fourier-Feynman transform on the function space C_a,b[0, T]. The function space C_a,b[0, T] is induced by the generalized Brownian motion process with mean function a(t) and variance function b(t). To do this, we study the generalized Fourier-Feynman transform associated with the Gaussian process Ƶ_k of exponential-type functionals. We then establish that a composition of the Ƶ_k-generalized Fourier-Feynman transforms acts like an inverse generalized Fourier-Feynman transform.

• Journal for History of Mathematics
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• v.20 no.3
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• pp.73-90
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• 2007

In this paper, we define a generalized Feynman integral and a generalized Fourier-Feynman transform on function space induced by generalized Brownian motion process. We then give existence theorems and several properties for these concepts. Finally we investigate relationships of the Fourier transform and the generalized Fourier-Feynman transform.

• Communications of the Korean Mathematical Society
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• v.11 no.3
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• pp.707-723
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• 1996

We first define the concept of the generalized analytic Fourier-Feynman transforms of a class of functionals on function space induced by a generalized Brownian motion process and study of functionals which plays on important role in physical problem of the form $ F(x) = {\int^{T}_{0} f(t, x(t))dt} $ where f is a complex-valued function on $[0, T] \times R$. We next show that the generalized analytic Fourier-Feynman transform of the convolution product is a product of generalized analytic Fourier-Feynman transform of functionals on functin space.