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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Classes of $L^{1}$-convergence of Fourier and Fourier-Stieltjes series


Author: Časlav V. Stanojević
Journal: Proc. Amer. Math. Soc. 82 (1981), 209-215
MSC: Primary 42A32; Secondary 42A20
DOI: https://doi.org/10.1090/S0002-9939-1981-0609653-4
MathSciNet review: 609653
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Abstract: It is shown that the Fomin class ${\mathcal {F}_p}(1 < p \leqslant 2)$ is a subclass of $\mathcal {C} \cap \mathcal {B}\mathcal {V}$, where $\mathcal {C}$ is the Garrett-Stanojević class and $\mathcal {B}\mathcal {V}$ the class of sequences of bounded variation. Wider classes of Fourier and Fourier-Stieltjes series are found for which ${a_n}\;{\text {lg}}\;n = o(1),n \to \infty$, is a necessary and sufficient condition for ${L^1}$-convergence. For cosine series with coefficients in $\mathcal {B}\mathcal {V}$ and $n\Delta {a_n} = O(1)$, $n \to \infty$, necessary and sufficient integrability conditions are obtained.


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Keywords: <IMG WIDTH="28" HEIGHT="23" ALIGN="BOTTOM" BORDER="0" SRC="images/img1.gif" ALT="${L^1}$">-convergence of Fourier series and Fourier-Stieltjes series, integrability of cosine series
Article copyright: © Copyright 1981 American Mathematical Society