• 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 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.

• 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.

• Journal of Integrative 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 $\{{\alpha}_{\kappa}\}{\in}{\Gamma}$.

• 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.

• 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.

• Kyungpook Mathematical Journal
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• v.45 no.1
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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.