• Communications of the Korean Mathematical Society
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• v.24 no.3
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• pp.397-413
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• 2009

In this paper, for functionals of a generalized Brownian motion process, we show that the generalized Fourier-Feynman transform of the convolution product is a product of multiple transforms and that the conditional generalized Fourier-Feynman transform of the conditional convolution product is a product of multiple conditional transforms. This allows us to compute the (conditional) transform of the (conditional) convolution product without computing the (conditional) convolution product.

• 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.

• Journal of the Korean Mathematical Society
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• v.38 no.1
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• pp.61-76
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• 2001

In this paper we define the concept of a conditional Fourier-Feynman transform and a conditional convolution product and obtain several interesting relationships between them. In particular we show that the conditional transform of the conditional convolution product is the product of conditional transforms, and that the conditional convolution product of conditional transforms is the conditional transform of the product of the functionals.

• Bulletin of the Korean Mathematical Society
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• v.54 no.2
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• pp.687-704
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• 2017

Using a simple formula for conditional expectations over an analogue of Wiener space, we calculate a generalized analytic conditional Fourier-Feynman transform and convolution product of generalized cylinder functions which play important roles in Feynman integration theories and quantum mechanics. We then investigate their relationships, that is, the conditional Fourier-Feynman transform of the convolution product can be expressed in terms of the product of the conditional FourierFeynman transforms of each function. Finally we establish change of scale formulas for the generalized analytic conditional Fourier-Feynman transform and the conditional convolution product. In this evaluation formulas and change of scale formulas we use multivariate normal distributions so that the orthonormalization process of projection vectors which are essential to establish the conditional expectations, can be removed in the existing conditional Fourier-Feynman transforms, conditional convolution products and change of scale formulas.

• 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.

• Korean Journal of Mathematics
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• v.16 no.3
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• pp.335-348
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• 2008

We define Fourier-Yeh-Feynman transform and convolution product on the Yeh-Wiener space, and establish the existence of Fourier-Yeh-Feynman transform and convolution product for functionals in a Banach algebra $\mathcal{S}(Q)$. Also we obtain Parseval's relation for those functionals.

• 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.

• East Asian mathematical journal
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• v.31 no.1
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• pp.135-144
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• 2015

In this paper we establish several formulas for multiple integral transform of functionals defined on abstract Wiener space. We then use the these results to establish several basic formulas involving multiple convolution products.

• Bulletin of the Korean Mathematical Society
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• v.48 no.5
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• pp.1003-1014
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• 2011

Motivated by the convolution product of white noise functionals, we introduce a new notion of convolution products of white noise operators. Then we study several interesting relations between the convolution products and the quantum generalized Fourier-Mehler transforms, and study a quantum-classical correspondence.

• East Asian mathematical journal
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• v.29 no.5
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• pp.467-479
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• 2013

We develop a Fourier-Feynman theory for Fourier-type functionals ${\Delta}^kF$ and $\widehat{{\Delta}^kF}$ on Wiener space. We show that Fourier-Feynman transform and convolution of Fourier-type functionals exist. We also show that the Fourier-Feynman transform of the convolution product of Fourier-type functionals is a product of Fourier-Feynman transforms of each functionals.