• Communications of the Korean Mathematical Society
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• v.34 no.2
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• pp.507-522
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• 2019

The main aim of this present paper is to present a new extension of the fractional derivative operator by using the extension of beta function recently defined by Shadab et al. [19]. Moreover, we establish some results related to the newly defined modified fractional derivative operator such as Mellin transform and relations to extended hypergeometric and Appell's function via generating functions.

• Journal of applied mathematics & informatics
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• v.37 no.1_2
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• pp.13-21
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• 2019

In the present research paper, we introduce a further extension of Hurwitz-Lerch zeta function by using the generalized extended Beta function defined by Parmar et al.. We investigate its integral representations, Mellin transform, generating functions and differential formula. In view of diverse applications of the Hurwitz-Lerch Zeta functions, the results presented here may be potentially useful in some related research areas.

• Journal of the Korea Institute of Information and Communication Engineering
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• v.23 no.3
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• pp.291-298
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• 2019

In this paper, we propose a Fresnel transform method using various wavelet functions to efficiently decompose digital holograms. After implementing the proposed wavelet function-based Fresnelet transforms, we apply it to the digital hologram and analyze the energy characteristics of the coefficients. The implemented wavelet transform-based Fresnelet transform is well suited for reconstructing and processing holograms which are optically obtained or generated by computer-generated hologram technique. After analyzing the characteristics of the spline function, we discuss wavelet multiresolution analysis method based on it. Through this process, we proposed a transform tool that can effectively decompose fringe patterns generated by optical interference phenomena. We implement Fresnelet transform based on wavelet function with various decomposition properties and show the results of decomposing fringe pattern using it. The results show that the energy distribution of the coefficients is significantly different depending on whether the random phase is included or not.

• Proceedings of the Korean Institute of Information and Commucation Sciences Conference
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• 2008.05a
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• pp.155-158
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• 2008

In this paper, we propose a novel blind watermarking Method using 2D CAT (Two dimensional cellular automata transform). In our scheme, firstly, we obtain the gateway values to generate a dual-state, dual-coefficients basis function. Secondly, the basis function transforms images into cellular automata space. Lastly, we use the cellular automata transform coefficients to embed random noise watermark in the cover images. The proposed scheme allows only one 2D CAT basis function per gateway value. Since there are $2^{96}$ possible gateway values, better security is guaranteed. Moreover, the new method not only verifies higher fidelity than the existing method but also stronger stability on JPEG lossy compression, filtering, sharpening and noise through tests for robustness.

• 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.

• Honam Mathematical Journal
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• v.38 no.4
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• pp.739-752
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• 2016

The main aim of this paper is to introduce an extension of the generalized ${\tau}$-Gauss hypergeometric function $_rF^{\tau}_s(z)$ and investigate various properties of the new function such as integral representations, derivative formulas, Laplace transform, Mellin trans-form and fractional calculus operators. Some of the interesting special cases of our main results have been discussed.

• Journal of the Korean Mathematical Society
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• v.59 no.5
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• pp.927-944
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• 2022

Let $\vec{p}{\in}(0,\;1]^n$ be an n-dimensional vector and A a dilation. Let $H^{\vec{p}}_A(\mathbb{R}^n)$ denote the anisotropic mixed-norm Hardy space defined via the radial maximal function. Using the known atomic characterization of $H^{\vec{p}}_A(\mathbb{R}^n)$ and establishing a uniform estimate for corresponding atoms, the authors prove that the Fourier transform of $f{\in}H^{\vec{p}}_A(\mathbb{R}^n)$ coincides with a continuous function F on ℝⁿ in the sense of tempered distributions. Moreover, the function F can be controlled pointwisely by the product of the Hardy space norm of f and a step function with respect to the transpose matrix of A. As applications, the authors obtain a higher order of convergence for the function F at the origin, and an analogue of Hardy-Littlewood inequalities in the present setting of $H^{\vec{p}}_A(\mathbb{R}^n)$.

• Journal of the Korean Mathematical Society
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• v.51 no.6
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• pp.1321-1322
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• 2014

• Communications of the Korean Mathematical Society
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• v.33 no.2
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• pp.515-525
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• 2018

In this paper, we investigate a modified $G^2$ transform on a class of Boehmians. We prove the axioms which are necessary for establishing the $G^2$ class of Boehmians. Addition, scalar multiplication, convolution, differentiation and convergence in the derived spaces have been defined. The extended $G^2$ transform of a Boehmian is given as a one-to-one onto mapping that is continuous with respect to certain convergence in the defined spaces. The inverse problem is also discussed.

• Kyungpook Mathematical Journal
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• v.46 no.1
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• pp.49-56
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• 2006

The aim of the present work is to obtain the inverse Laplace transform of the product of the factors of the type $s^{-\rho}\prod\limit_{i=1}^{\tau}(s^{li}+{\alpha}_i)^{-{\sigma}i}$, a general class of polynomials an the multivariable H-function. The polynomials and the functions involved in our main formula as well as their arguments are quite general in nature. On account of the general nature of our main findings, the inverse Laplace transform of the product of a large variety of polynomials and numerous simple special functions involving one or more variables can be obtained as simple special cases of our main result. We give here exact references to the results of seven research papers that follow as simple special cases of our main result.