• 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.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.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.

• Kyungpook Mathematical Journal
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• v.54 no.1
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• pp.31-41
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• 2014

In this paper criterion for $L_1$-convergence of a certain cosine sums with quasi hyper-convex coefficients is obtained.

• Kyungpook Mathematical Journal
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• v.47 no.3
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• pp.323-328
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• 2007

In this paper, we extend the result of Ram [3] and also study the $L^1$-convergence of the $r^{th}$ derivative of cosine series.

• 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 $\{{\alpha}_{\kappa}\}{\in}{\Gamma}$.

• Bulletin of the Korean Mathematical Society
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• v.44 no.1
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• pp.125-130
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• 2007

We propose a simple and intuitive method to derive the exact convergence rate of global $L_{2}-norm$ error for strong numerical approximation of stochastic differential equations the result of which has been reported by Hofmann and $M{\"u}ller-Gronbach\;(2004)$. We conclude that any strong numerical scheme of order ${\gamma}\;>\;1/2$ has the same optimal convergence rate for this error. The method clearly reveals the structure of global $L_{2}-norm$ error and is similarly applicable for evaluating the convergence rate of global uniform approximations.

• 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.

• Kyungpook Mathematical Journal
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• v.46 no.1
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• pp.111-118
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• 2006

The present paper investigates the weighted $L^1$-convergence of Gr$\ddot{u}$nwald interpolatory operators based on the zeros of the second Chebyshev polynomials $U_n(x)=\frac{sin(n+1)\theta}{sin\theta}$. The approximation rate is sharp.

• International Journal of Fuzzy Logic and Intelligent Systems
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• v.9 no.1
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• pp.59-64
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• 2009

In this paper, we study the relations between L-fuzzy topologies and L-filters on a strictly two-sided, commutative quantale lattice L. We define an L-fuzzy neighborhood filter and introduce the notion of L-filter convergence in L-fuzzy topological spaces.