• East Asian mathematical journal
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• v.14 no.1
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• pp.157-163
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• 1998

The aim of this note is to prove Vandermonde's convolution theorem by using the theory of hypergeometric series as suggested in literature which does not seem to be easy to justify it. We also provide an interesting identity and its application.

• The Transactions of the Korean Institute of Electrical Engineers D
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• v.52 no.1
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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.

• Journal of the Korean Mathematical Society
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• v.57 no.3
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• pp.655-668
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• 2020

In this paper, we establish the generalized Cameron-Storvick type theorem on function space. We then give relationships involving the generalized Cameron-Storvick type theorem, modified generalized integral transform and modified convolution product. A motivation of studying the generalized Cameron-Storvick type theorem is to generalize formulas and results with respect to the modified generalized integral transform on function space. From the some theories and formulas in the functional analysis, we can obtain some formulas with respect to the translation theorem of exponential functionals.

• Journal of applied mathematics & informatics
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• v.16 no.1_2
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• pp.633-641
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• 2004

In this paper we deal with functions in higher dimension. We provide several convergence theorem for approximation by convolution type delta sequence. We also give sufficient and necessary condition for Gibbs phenomenon to exist.

• 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.

• Kyungpook Mathematical Journal
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• v.58 no.4
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• pp.599-613
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• 2018

We study a generalization of the classical Pentagonal Number Theorem and its applications. We derive new identities for certain infinite series, recurrence relations and convolution sums for certain restricted partitions and divisor sums. We also derive new identities for Bell polynomials.

• Kyungpook Mathematical Journal
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• v.51 no.3
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• pp.251-260
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• 2011

The aim of this paper is to obtain a solution of a certain multidimensional convolution integral equation of Fredholm type whose kernel involves a generalized polynomial set. A number of results follow as special cases from the main theorem by specifying the parameters of the generalized polynomial set.

• Honam Mathematical Journal
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• v.35 no.1
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• pp.1-15
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• 2013

In this paper, we study a sheaf-theoretic analysis of the convolution algebra on quiver varieties. As by-products, we reinterpret the results of H. Nakajima. We also produce a refined form of the BBD decomposition theorem for quiver varieties. Finally, we study a construction of highest weight modules through constructible functions.

• Journal of the Korean Mathematical Society
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• v.32 no.3
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• pp.615-623
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• 1995

Consider a function m defined in $R^n$ and the associated convolution operator T given by the Fourier transform $\hat{T} = m\hat{f}$.

• Kyungpook Mathematical Journal
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• v.34 no.1
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• pp.7-10
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• 1994