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Classes of $ L\sp{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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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1981-0609653-4
Keywords: $ {L^1}$-convergence of Fourier series and Fourier-Stieltjes series, integrability of cosine series
Article copyright: © Copyright 1981 American Mathematical Society

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