• Journal of applied mathematics & informatics
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• 제31권1_2호
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• pp.263-276
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• 2013

We discuss the effects of the ${\delta}^2$ and Lubkin acceleration methods on the partial sums of Fourier Series. We construct continuous, even H$\ddot{o}$lder continuous functions, for which these acceleration methods fail to give convergence. The constructed functions include some interesting trigonometric series whose properties were investigated by Hardy and Littlewood.

• 한국수학사학회지
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• 제30권4호
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• pp.233-246
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• 2017

$Ces{\grave{a}}ro$ summability is a generalized convergence criterion for infinite series. We have investigated the classical results of summability for Fourier series from 1897 to 1957. In this paper, we are concerned with the summability and summation methods for Fourier Series from 1960 to 2010. Many authors have studied the subject during this period. Especially, G.M. Petersen,$K{\hat{o}}si$ Kanno, S.R. Sinha, Fu Cheng Hsiang, Prem Chandra, G. D. Dikshit, B. E. Rhoades and others had studied neoclassical results on the summability of Fourier series from 1960 to 1989. We investigate the results on the summability for Fourier series from 1990 to 2010 in section 3. In conclusion, we present the research minor lineage on summability for Fourier series from 1960 to 2010. $H{\ddot{u}}seyin$ Bor is the earliest researcher on ${\mid}{\bar{N}},p_n{\mid}_k$-summability. Thus we consider his research results and achievements on ${\mid}{\bar{N}},p_n{\mid}_k$-summability and ${\mid}{\bar{N}},p_n,{\gamma}{\mid}_k$-summability.

• 전기전자학회논문지
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• 제18권4호
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• pp.463-470
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• 2014

In stratified flows of two immiscible liquids, due to the vibration in a pipe, the shape of the interface is not always periodic and it causes the different end points of the interfacial boundary. In this case the performance is not good. To solve this, in this paper, the hybrid-type Fourier series is proposed, which consists of both the polynomial and the trigonometric terms. Under the stationary interfacial boundary during acquiring a full set of voltage data, the performance of the proposed method is evaluated through the numerical experiments. The results show that the proposed method performs better than the conventional Fourier series in estimating the interfacial boundary.

• Journal of the Optical Society of Korea
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• 제16권3호
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• pp.196-202
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• 2012

We observe that the transmission and reflection efficiencies of a one-dimensional metallic grating under transverse-magnetic illumination calculated using the Fourier modal method (FMM) with the Fourier factorization rules have peculiar fluctuations, albeit small in magnitude, as the number of field harmonics increases. It is shown that when the number of Fourier terms for the electromagnetic field is increased from that in the conventional FMM, the fluctuations due to non-convergent highly evanescent eigenmodes can be eliminated. Our examination reveals that the fluctuations originate from the Gibbs phenomenon inherent in the Fourier-series representation of a permittivity function with discontinuities, and from non-convergence of highly evanescent internal Bloch eigenmodes.

• 한국수학사학회지
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• 제30권6호
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• pp.353-365
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• 2017

In general, there is summability among the mathematical tools that are the criterion for the convergence of infinite series. Many authors have studied on the summability of infinite series, the summability of Fourier series and the summability factors. Especially, $H{\ddot{u}}seyin$ Bor had published his important results on these topics from the beginning of 1980 to the end of 1990. In this paper, we investigate the minor academic genealogy of teachers and pupils from Fourier to $H{\ddot{u}}seyin$ Bor in section 2. We introduce the $H{\ddot{u}}seyin$ Bor's major results of the summability for infinite series from 1983 to 1997 in section 3. In conclusion, we summarize his research characteristics and significance on the summability of infinite series. Also, we present the diagrams of $H{\ddot{u}}seyin$ Bor's minor academic genealogy from Fourier to $H{\ddot{u}}seyin$ Bor and minor research lineage on the summability of infinite series.

• 한국해양공학회지
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• 제37권2호
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• pp.49-57
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• 2023

Since Chappelear developed a Fourier approximation method, considerable research efforts have been made. On the other hand, Fourier approximations are unsuitable for deep water waves. The purpose of this study is to provide a Fourier approximation suitable even for deep water waves and a numerical method to determine the Fourier coefficients and the wave properties. In addition, the convergence of the solution was tested in terms of its order. This paper presents a velocity potential satisfying the Laplace equation and the bottom boundary condition (BBC) with a truncated Fourier series. Two wave profiles were derived by applying the potential to the kinematic free surface boundary condition (KFSBC) and the dynamic free surface boundary condition (DFSBC). A set of nonlinear equations was represented to determine the Fourier coefficients, which were derived so that the two profiles are identical at specified phases. The set of equations was solved using Newton's method. This study proved that there is a limit to the series order, i.e., the maximum series order is N=12, and that there is a height limitation of this method which is slightly lower than the Michell theory. The reason why the other Fourier approximations are not suitable for deep water waves is discussed.

• 전자공학회논문지B
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• 제30B권4호
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• pp.67-75
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• 1993

The Fourier series are employed to approximate the input/output(I/O) characteristics of a dynamic system and, based on the approximation, a new learing control algorithm is proposed in order to find iteratively the control input for tracking a desired trajectory. The use of the Fourier approximation of I/O renders at least a couple of useful consequences: the frequency characteristics of the system can be used in the controller design and the reconstruction of the system states is not required. The convergence condition of the proposed algorithm is provided and the existence and uniqueness of the desired control input is discussed. The effectiveness of the proposed algorithm is illustrated by computer simulation for a robot trajectory tracking. It is shown that, by adding feedback term in learning control algorithm, robustness and convergence speed can be improved.

• 대한수학회보
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• 제21권2호
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• pp.138-138
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• 1984

• 한국수학사학회지
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• 제19권2호
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• pp.115-124
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• 2006

유계변동(bounded variation)과 관련된 Daniel Waterman의 30여 년간의 연구에 대하여 조사를 하였다.

• Kyungpook Mathematical Journal
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• 제45권1호
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• pp.73-85
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• 2005

In this paper a criterion for $L^1-convergence$ of a new modified sine sums is obtained by using $Ces{\grave{a}}ro$ means of integral and non-integral orders.