• 대한수학회지
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• 제42권3호
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• pp.599-619
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• 2005

We define the convolutions of Fourier hyperfunctions and show that every strongly decreasing Fourier hyperfunction is a convolutor for the space of Fourier hyperfunctions and the converse is true. Also we show that there are no differential operator with constant coefficients which have a fundamental solution in the space of strongly decreasing Fourier hyperfunctions. Lastly we show that the space of multipliers for the space of Fourier hyperfunctions consists of analytic functions extended to any strip in $\mathbb{C}^n$ which are estimated with a special exponential function exp$(\mu|\chi|)$.

• 한국광학회지
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• 제6권3호
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• pp.245-256
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• 1995

Fourier 면환을 이용하여 불균일 굴절률 박막의 rugate 필터를 설계하였으며 rugate 필터의 반사율, 대역폭, 광학 두께, Q 함수 등을 변화시키며 Fourier 변환의 여러 가지 특성을 조사하였다. 주어진 단선 및 이중 rugate 필터의 과녁 스펙트럼에 불균일 굴절률 박막의 스펙트럼을 맞추기 위하여 merit 함수를 사용하였으며 merit 값이 최소가 되도록 Q 함수를 반복계산하여 수정하였다. Sossi, Bovard, Fabricius가 각각 유도한 세 종류의 Q함수를 반복계산 횟수, merit 함수의 값, 최적 광학두께 등의 관점에서 비교하였다. 반사율이 높은 rugate 필터 설계에는 반복계산 수정 후 반사율이 과녁스펙트럼에 가까운 Bovard와 Fabricius의 Q함수가 적당하며, 광학 두께는 최소 광학두께만 넘으면 반복계산 수정과정을 이용하여 과녁반사율을 맞출 수 있으므로 반사대역폭이 허용하는 광학두께로 결정하면 될 것이다.

• 대한수학회논문집
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• 제36권3호
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• pp.485-494
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• 2021

In this article, we prove the existence of a solution to some classes of integral equations of generalized convolution type generated by the Kontorovich-Lebedev (K) transform, the Laplace (𝓛) transform and the Fourier (F) transform in some appropriate function spaces and represent it in a closed form.

• 한국수학교육학회지시리즈B:순수및응용수학
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• 제19권2호
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• pp.87-102
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• 2012

In this paper, we consider the Fourier-type functionals introduced in [16]. We then establish the analytic Feynman integral for the Fourier-type functionals. Further, we get a series expansion of the analytic Feynman integral for the Fourier-type functional $[{\Delta}^kF]^{\^}$. We conclude by applying our series expansion to several interesting functionals.

• East Asian mathematical journal
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• 제26권1호
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• pp.105-113
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• 2010

In the spectral analysis, Fourier coeffcients are very important to give informations for the original signal f on a finite domain, because they recover f. Also Fourier analysis has extension to wavelet analysis for the whole space R. Various kinds of reconstruction theorems are main subject to analyze signal function f in the field of wavelet analysis. In this paper, we will present a new reconstruction theorem of functions in $L^1(R)$ using a nonharmonic Fourier series. When we construct this series, we have used dyadic subdivision schemes.

• 대한의용생체공학회:의공학회지
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• 제10권3호
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• pp.323-330
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• 1989

Large bias errors which occur during a numerical evaluation of the Rayleigh's integral is not due to the replicated source problem but due to the coincidence of singularities of the Green's function and the sampling points in Fourier domain. We found that there is no replicated source problem in evaluating the Rayleigh's integral numerically by the reason of the periodic assumption of the input sequence in Dn or by the periodic sampling of the Green's function in the Fourier domain. The wrap around error is not due to an overlap of the individual adjacent sources but berallse of the undersampling of the Green's function in the frequency domain. The replicated and overlApped one is inverse Fourier transformed Green's function rather than the source function.

• 대한수학회지
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• 제59권5호
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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 $H^{\vec{p}}_A(\mathbb{R}^n)$.

• 대한수학회지
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• 제32권4호
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• pp.785-802
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• 1995

It is well known that every periodic function $f \in L^p([0,2\pi]), p > 1$, can be represented by a convergent trigonometric series called the Fourier series of f. Uniqueness of the representing series is very important, and we know that the Fourier series of a periodic function $f \in L^p([0,2\pi])$ is unique.

• 대한수학회보
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• 제60권5호
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• pp.1221-1235
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• 2023

In this paper, we use an infinite dimensional conditioning function to define a conditional Fourier-Feynman transform (CFFT) and a conditional convolution product (CCP) on the Wiener space. We establish the existences of the CFFT and the CCP for bounded functions which form a Banach algebra. We then provide fundamental relationships between the CFFTs and the CCPs.

• 한국수학교육학회지시리즈B:순수및응용수학
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• 제30권2호
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• pp.155-167
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• 2023

In this paper, we use a vector-valued conditioning function to define a conditional Fourier-Feynman transform (CFFT) and a conditional convolution product (CCP) on the Wiener space. We establish the existences of the CFFT and the CCP for bounded functionals which form a Banach algebra. We then provide fundamental relationships between the CFFTs and the CCPs.