• 한국수학사학회지
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• 제27권1호
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• pp.81-93
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• 2014

The $L^1$-convergence of Fourier series problems through additional assumptions for Fourier coefficients were presented by W. H. Young in 1913. We say that they are the classical results. Using modified trigonometric series is the convenience method to study the $L^1$-convergence of Fourier series problems. they are called the neoclassical results. This study concerns with the $L^1$-convergence of Fourier series. We introduce the classical and neoclassical results of $L^1$-convergence sequentially. In particular, we investigate $L^1$-convergence results focused on the results of Bhatia's studies. In conclusion, we present the research minor lineage of Bhatia's studies and compare the classes of $L^1$-convergence mutually.

• 소음진동
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• 제6권6호
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• pp.771-779
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• 1996

In this paper, double Fourier sine series is used as a modal displacement functions of a rectangular plate and applied to the free vibration analysis of a rectangular plate under various boundary conditions. The method of stationary potential energy is used to obtain the modal displacements of a plate. To enhance the flexibility of the double Fourier sine series, Lagrangian multipliers are utilized to match the geometric boundary conditions, and Stokes' transformation is used to handle the displacements that are not satisfied by the double Fourier sine series. The frequency parameters and mode shapes obtained by the present method are compared with those obtained by MSC/NASTRAN and other analysis.

• 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.

• Journal of applied mathematics & informatics
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• 제26권5_6호
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• pp.1101-1121
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• 2008

One proposes an approach to fractional Fourier's transform, or Fourier's transform of fractional order, which applies to functions which are fractional differentiable but are not necessarily differentiable, in such a manner that they cannot be analyzed by using the so-called Caputo-Djrbashian fractional derivative. Firstly, as a preliminary, one defines fractional sine and cosine functions, therefore one obtains Fourier's series of fractional order. Then one defines the fractional Fourier's transform. The main properties of this fractal transformation are exhibited, the Parseval equation is obtained as well as the fractional Fourier inversion theorem. The prospect of application for this new tool is the spectral density analysis of signals, in signal processing, and the analysis of some partial differential equations of fractional order.

• Journal of applied mathematics & informatics
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• 제41권5호
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• pp.1057-1069
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• 2023

In this paper, we have determined the degree of approximation of function belonging of Lipschitz class by using Deferred-Generalized Nörlund (D^𝛾_𝛽.N_pq) means of Fourier series and conjugate series of Fourier series, where {p_n} and {q_n} is a non-increasing sequence. So that results of DEGER and BAYINDIR [23] become special cases of our results.

• 대한수학회보
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• 제57권2호
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• pp.481-488
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• 2020

In this paper we give an exact formula for the Fourier coefficients of the Jacobi-Eisenstein series of square free discriminant lattice index. For a special case the discriminant of lattice is prime we show that the Jacobi-Eisenstein series corresponds to a well known Eisenstein series of modular forms.

• 호남수학학술지
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• 제43권2호
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• pp.259-267
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• 2021

Given 𝛽 > 1, we consider real numbers whose 𝛽-expansions are Sturmian words. When the slope of Sturmian words varies, their behaviors have been well studied from analytical point of view. The regularity enables us to find the Fourier series expansion, while the singularity at rational slopes yields a new kind of trigonometric series representing 𝜋.

• 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.

• 통합자연과학논문집
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• 제1권3호
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• pp.216-220
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• 2008

In general the Banach space $L^1(T^1)$ doesn't admit convergence in norm. Thus the convergence in norm of the partial sums can not be characterized in terms of Fourier coefficients without additional assumptions about the sequence$\{^{\^}f(\xi)\}$. The problem of $L^1(T^1)$-convergence consists of finding the properties of Fourier coefficients such that the necessary and sufficient condition for (1.2) and (1.3). This paper showed that let $\{{\alpha}_{\kappa}\}{\in}BV$ and ${\xi}{\Delta}a_{\xi}=o(1),\;{\xi}{\rightarrow}{\infty}$. Then (1.1) is a Fourier series if and only if $\{{\alpha}_{\kappa}\}{\in}{\Gamma}$.

• 한국수학사학회지
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• 제22권1호
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• pp.25-40
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• 2009

본 논문에서는 푸리에의 생애와 18세기 말 그의 스승과 19세기부터 20세기까지는 그의 제자와 후학들의 소계보를 살펴보고 특히, 비교적 덜 접근된 러시아 수학자들의 푸리에 급수의 $L^1$-수렴성에 대한 연구결과들 중 푸리에 계수 성질을 이용한 푸리에 급수 수렴성에 대해 매우 의미 있는 연구를 이룩한 콜모고로프, 테라코브스키의 연구결과에 관심을 갖고 이들의 연구 결과를 비교하여 조사하였다.