• Title/Summary/Keyword: convergence of Fourier series

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On Classical Studies for Summability and Convergence of Double Fourier Series (이중 푸리에 급수의 총합가능성과 수렴성에 대한 고전적인 연구들에 관하여)

  • Lee, Jung Oh
    • 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.

Partial Sum of Fourier series, the Reinterpret of $L^1$-Convergence Results using Fourier coefficients and theirs Minor Lineage (푸리에 급수의 부분합, 푸리에 계수를 이용한 $L^1$-수렴성 결과들의 재해석과 그 소계보)

  • Lee, Jung-Oh
    • 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.

CONVERGENCE RATE OF HYBRID SAMPLING SERIES ASSOCIATED WITH WAVELETS

  • Shim, Hong-Tae;Kwon, Joong-Sung
    • Journal of applied mathematics & informatics
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    • v.14 no.1_2
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    • pp.267-275
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    • 2004
  • While the convergence of the classical Fourier series has been well known, the rate of its convergence is not well acknowledged. The results regarding the rate of convergence of the Fourier series and wavelet expansions can be found in the book of Walter[5]. In this paper, we give the rate of convergence of hybrid sampling series associated with orthogonal wavelets.

A Brief Study on Bhatia's Research of L1-Convergence (바티의 L1-수렴성 연구에 관한 소고)

  • Lee, Jung Oh
    • Journal for History of Mathematics
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    • v.27 no.1
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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.

On Lp(T2)-Convergence and Móricz (Lp(T2)-수렴성과 모리츠에 관하여)

  • LEE, Jung Oh
    • 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.

A Brief Study on Stanojevic's Works on the $\mathfrak{L}^1$-Convergence (Stanojevic의 푸리에 급수의 $\mathfrak{L}^1$-수렴성 연구의 소 계보 고찰)

  • Lee, Jung Oh
    • Journal for History of Mathematics
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    • v.26 no.2_3
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    • pp.163-176
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    • 2013
  • This study concerns Stanojevic's academic works on the $\mathfrak{L}^1$-convergence of Fourier series from 1973 to 2002. We review his academic works. Also, we briefly investigate a simple academic lineage for the researchers of $\mathfrak{L}^1$-convergence of Fourier series until 2012. First, we introduce the classical lineage of the researchers for $\mathfrak{L}^1$-convergence Fourier series in section 2. Second, we investigate the backgrounds of Stanojevic's study at Belgrade University and University of Missouri-Rolla respectively. Finally, we compare and consider the $\mathfrak{L}^1$-convergence theorems of Stanojevic's results from 1973 to 2002 successively. In addition, we compose a the simple lineage of $\mathfrak{L}^1$-convergence of Fourier series from 1973 to 2012.

On $L^1(T^1)$ - Convergence of Fourier Series with BV - Class Coefficients (BV - 족 계수를 갖는 푸리에 급수의 $L^1(T^1)$ - 수렴성에 관하여)

  • Lee, Jung-Oh
    • Journal of the Chosun Natural Science
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    • v.1 no.3
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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 $\{{\alpha}_{\kappa}\}{\in}{\Gamma}$.

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The Life of Fourier, The minor Lineage of His Younger Scholars and a Theorem of Telyakovskii on $L^1$-Convergence (푸리에 일생, 푸리에 후학의 소계보와 $L^1$-수렴성에 관한 테라코브스키의 정리)

  • Lee, Jung-Oh
    • Journal for History of Mathematics
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    • v.22 no.1
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    • pp.25-40
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    • 2009
  • This study concerns with John B. Fourier' s life, his teachers, his younger scholars and the $L^1$-convergence of Fourier series. First, we introduce the correlation between the French Revolution and Fourier who is significant in the history of mathematics. Second, we investigate Fourier' s teachers, students and a minor lineage of his younger scholars from 19th century to 20th century. Finally, we compare the theorem of Telyakovskii with the theorem of kolmogorov on $L^1$-convergence of Fourier series.

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ON THE ABSOLUTE CONVERGENCE OF LACUNARY VECTOR VALUED FOURIER COEFFICIENTS SERIES

  • Rashwan, R.A.
    • Kyungpook Mathematical Journal
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    • v.27 no.2
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    • pp.173-179
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    • 1987
  • In this article the absolute convergence of lacunary Fourier Coefficients Series is studied for Hilbert space valued functions. The considered functions arc assumed to be of either the modulus of continuity or the modulus of smoothness of order l which are considered only at a fixed point in [$-{\pi},{\pi}$]. On the other hand for values in weakly sequentially complete Banach space, the lacunary Fourier coefficients series is strongly unconditionally convergent. The results obtained here are a kind of a generalization of the results due to Kandil [4].

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