• Journal of Advanced Marine Engineering and Technology
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• v.8 no.1
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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.

• Journal for History of Mathematics
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• v.27 no.4
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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.

• Journal for History of Mathematics
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• v.23 no.1
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• pp.53-66
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• 2010

This study concerns with partial sum of Fourier series, Fourier coefficients and the $L^1$-convergence of Fourier series. First, we introduce the $L^1$-convergence results. We consider equivalence relations of the partial sum of Fourier series from the early 20th century until the middle of. Second, we investigate the minor lineage of $L^1$-convergence theorem from W. H. Young to G. A. Fomin. Finally, we compare and reinterpret the $L^1$-convergence theorems.

• Journal of applied mathematics & informatics
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• v.17 no.1_2_3
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• pp.719-736
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• 2005

An understanding of Fourier series and their generalization is important for physics and engineering students, as much for mathematical and physical insight as for applications. Students are usually confused by the so-called Gibbs' phenomenon, an overshoot between a discontinuous function and its approximation by a Fourier series as the number of terms in the series becomes indefinitely large. In this paper we give short story of Gibbs phenomenon in chronological order.

• Journal for History of Mathematics
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• v.30 no.1
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• pp.17-29
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• 2017

This paper is concerned with the $Ces{\grave{a}}ro$ summability of Fourier series. Many authors have studied on the summability of Fourier series up to now. Also, G. H. Hardy and J. E. Littlewood [5], Gaylord M. Merriman [18], L. S. Bosanquet [1], Fu Traing Wang [24] and others had studied the $Ces{\grave{a}}ro$ summability of Fourier series until the first half of the 20th century. In the section 2, we reintroduce Ernesto $Ces{\grave{a}}ro^{\prime}s$ life and the meaning of mathematical history for $Ces{\grave{a}}ro^{\prime}s$ work. In the section 3, we investigate the classical results of summability for Fourier series from 1897 to the mid-twentieth century. In conclusion, we restate the important classical results of several theorems of $Ces{\grave{a}}ro$ summability for Fourier series. Also, we present the research minor lineage of $Ces{\grave{a}}ro$ summability for Fourier series.

• East Asian mathematical journal
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• v.26 no.1
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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.

• Journal for History of Mathematics
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• v.30 no.4
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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.

• Communications of the Korean Mathematical Society
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• v.26 no.1
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• pp.89-98
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• 2011

The Fourier series has a rapid oscillation near end points at jump discontinuity which is called the Gibbs phenomenon. There is an overshoot (or undershoot) of approximately 9% at jump discontinuity. In this paper, we prove that a bunch of series representations (certain nonharmonic Fourier series) give good approximations vanishing Gibbs phenomenon. Also we have an application for approximating some shape of upper part of a vehicle in a different way from the method of cubic splines and wavelets.

• The Pure and Applied Mathematics
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• v.19 no.2
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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.

• Journal for History of Mathematics
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• v.28 no.6
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• pp.321-332
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• 2015

This paper is concerned with the convergence of double trigonometric series and Fourier series. Since the beginning of the 20th century, many authors have studied on those series. Also, Ferenc $M{\acute{o}}ricz$ has studied the convergence of double trigonometric series and double Fourier series so far. We consider $L^p(T^2)$-convergence results focused on the Ferenc $M{\acute{o}}ricz^{\prime}s$ studies from the second half of the 20th century up to now. In section 2, we reintroduce some of Ferenc $M{\acute{o}}ricz^{\prime}s$ remarkable theorems. Also we investigate his several important results. In conclusion, we investigate his research trends and the simple minor genealogy from J. B. Joseph Fourier to Ferenc $M{\acute{o}}ricz$. In addition, we present the research minor lineage of his study on $L^p(T^2)$-convergence.