• 대한수학회논문집
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• 제24권3호
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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.

• 대한수학회지
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• 제38권1호
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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.

• 대한수학회보
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• 제41권1호
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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.

• 대한수학회보
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• 제54권2호
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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.

• Korean Journal of Mathematics
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• 제16권3호
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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.

• East Asian mathematical journal
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• 제29권5호
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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.

• Journal of applied mathematics & informatics
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• 제42권1호
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• pp.65-76
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• 2024

In this paper, the I-transform of the Mellin convolution type is presented. Based on the Mellin transform theory, a general integral transform of the Mellin convolution type is introduced. The generating operators for I-transform together with the corresponding operational relations are also presented.

• East Asian mathematical journal
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• 제31권1호
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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.

• Korean Journal of Mathematics
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• 제29권2호
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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.

• 대한전기학회논문지:시스템및제어부문D
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• 제52권1호
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• pp.51-54
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• 2003

The z-transform method is a basic mathematical tool in analyzing and designing digital signal processing systems for discrete input and output signals. There are may cases where the output signal is in the form of a discrete convolution sum of an input function and a designed digital processing algorithm function. It is well known that the z-transform of the convolution sum becomes the product of the two z-transforms of the input function and the digital processing function, whose proofs require the causality of the digital signal processing function in the almost all the available references. However, not all of the convolution sum functions are based on the causality. Many digital signal processing systems such as image processing system may depend not on the time information but on the spatial information, which has nothing to do with causality requirement. Thus, the application of the causality-based z-transform theorem on the convolution sum cannot be used without difficulty in this case. This paper proves the z-transform theorem on the discrete convolution sum without causality requirement, and make it possible for the theorem to be used in analysis and desing for any cases.