• 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.

• 자원환경지질
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• 제20권2호
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• pp.119-123
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• 1987

0차의 Hankel 변환은 Abel 변환과 Fourier 변환의 결합으로 나타낼 수 있다. 이 Abel-Fourier법은 샘플링 간격이 일정하지 않는 중합필터를 사용한 신속한 Abel 변환과 고속 Fourier 변환으로 작성될 때, 종래의 디지탈 선형필터법보다 계산시간면에서 유리하다. 그러나, Abel-Fourier법은 일반적으로 잘 설계된 디지탈 필터보다 정확하지는 않다. 전기탐사 문제에 이들 방법을 적용할 때, 디지탈 필터법이 Abel-Fourier 법보다 더 융통성이 많은 것으로 생각된다.

• Journal of Advanced Marine Engineering and Technology
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• 제8권1호
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• pp.100-111
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• 1984

The methods solving the 2nd order linear ordinary differential equations of the form y"+H(x)y'+G(x)y=P(x) using Fourier series are presented in this paper. These methods are applied to the differential equations of which the exact solutions are known, and the solutions by Fourier series are compared with the exact solutions. The main results obtained in these studies are summarized as follows; 1) The product and the quotient of two functions expressed in Fourier series can be expressed also in Fourier series and the relations between the Fourier coefficients of the series are obtained by multiplying term by term. 2) If the solution of the 2nd order lindar ordinary differential equation exists in a certain interval, the solution can be obtained using Fourier series and can be expressed in Fourier series. 3) The absolute errors of Fourier series solutions are generally less in the center of the interval than in the end of the interval. 4) The more terms are considered in Fourier series solutions, the less the absolute errors.rors.

• 대한수학회보
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• 제49권3호
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• pp.609-619
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• 2012

In this paper we define the Fourier-type functionals via the Fourier transform on Wiener space. We investigate some properties of the Fourier-type functionals. Finally, we establish integral transform of the Fourier-type functionals which also can be expressed by other Fourier-type functionals.

• Journal of the Optical Society of Korea
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• 제4권1호
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• pp.54-57
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• 2000

Introducing left and right fractional orders separately, we show that the cascading of optical extended fractional Fourier transform systems can be easily calculated. Through such calculations we study the properties of extended fractional Fourier transforms.

• 한국수학사학회지
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• 제27권4호
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• pp.285-297
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• 2014

G. H. Hardy laid the foundation of classical studies on double Fourier series at the beginning of the 20th century. In this paper we are concerned not only with Fourier series but more generally with trigonometric series. We consider Norlund means and Cesaro summation method for double Fourier Series. In section 2, we investigate the classical results on the summability and the convergence of double Fourier series from G. H. Hardy to P. Sjolin in the mid-20th century. This study concerns with the $L^1(T^2)$-convergence of double Fourier series fundamentally. In conclusion, there are the features of the classical results by comparing and reinterpreting the theorems about double Fourier series mutually.

• 대한수학회지
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• 제45권6호
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• pp.1785-1801
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• 2008

Noncommutative extensions of the Gross and Beltrami Laplacians, called the quantum Gross Laplacian and the quantum Beltrami Laplacian, resp., are introduced and their basic properties are studied. As noncommutative extensions of the Fourier-Gauss and Fourier-Mehler transforms, we introduce the quantum Fourier-Gauss and quantum Fourier- Mehler transforms. The infinitesimal generators of all differentiable one parameter groups induced by the quantum Fourier-Gauss transform are linear combinations of the quantum Gross Laplacian and quantum Beltrami Laplacian. A characterization of the quantum Fourier-Mehler transform is studied.

• 대한수학회지
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• 제43권5호
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• pp.967-990
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• 2006

In this paper, we evaluate first variations, conditional first variations and conditional Fourier-Feynman transforms of cylinder type functions over Wiener paths in abstract Wiener space and then, investigate relationships among first variation, conditional first variation, Fourier-Feynman transform and conditional Fourier-Feynman transform of those functions. Finally, we derive the conditional Fourier-Feynman transform for the product of cylinder type function which defines the functions in a Banach algebra introduced by Yoo, with n linear factors.

• Korean Journal of Mathematics
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• 제25권3호
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• pp.335-347
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• 2017

Time shifting and frequency shifting proprerties for the Fourier-Feynman transform of functionals in a generalized Fresnel class ${\mathcal{F}}_{A_1,A_2}$ are given. We discuss scaling and modulation proprerties for the Fourier-Feynman transform. These properties help us to obtain Fourier-Feynman transforms of new functionals from the Fourier-Feynman transforms of old functionals which we know their Fourier-Feynman transforms.

• 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$(\mu|\chi|)$.