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Proceedings of the American Mathematical Society

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Criteria for absolute convergence of Fourier series


Author: Nicolas Artémiadis
Journal: Proc. Amer. Math. Soc. 50 (1975), 179-183
MSC: Primary 42A28
DOI: https://doi.org/10.1090/S0002-9939-1975-0377398-1
MathSciNet review: 0377398
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Abstract: Let $ f \in {\mathbf{L}^1}({\mathbf{T}})$. Define $ {f_\alpha }$ by $ {f_\alpha }(x) = f(x + \alpha )$. Then the Fourier series of $ f$ is absolutely convergent if and only if there exists a Lebesgue point $ \alpha $ for $ f$ such that both sequences $ {\langle {({{\mathbf{R}}_{\text{e}}}{\hat f_\alpha }(n))^ - }\rangle _{n \in {... ...{\mathcal{I}_{\text{m}}}{\hat f_\alpha }(n))^ - }\rangle _{n \in {\mathbf{Z}}}}$ belong to $ {l^1}$. The theorem remains true if the sentence ``there exists a Lebesgue point $ \alpha $ for $ f$'' is replaced by ``there is $ \alpha \in {\mathbf{R}}$ such that $ f$ is essentially bounded in some neighborhood of $ \alpha $".


References [Enhancements On Off] (What's this?)

  • [1] Jean-Pierre Kahane, Séries de Fourier absolument convergentes, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 50, Springer-Verlag, Berlin-New York, 1970 (French). MR 0275043
  • [2] Walter Rudin, Fourier analysis on groups, Interscience Tracts in Pure and Applied Mathematics, No. 12, Interscience Publishers (a division of John Wiley and Sons), New York-London, 1962. MR 0152834

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DOI: https://doi.org/10.1090/S0002-9939-1975-0377398-1
Article copyright: © Copyright 1975 American Mathematical Society

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