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On a Tauberian theorem for the $ L\sp{1}$-convergence of Fourier sine series


Author: William O. Bray
Journal: Proc. Amer. Math. Soc. 88 (1983), 34-38
MSC: Primary 42A20
DOI: https://doi.org/10.1090/S0002-9939-1983-0691274-0
MathSciNet review: 691274
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Abstract: In a recent Tauberian theorem of Stanojević [3] for the $ {L^1}$-convergence of Fourier series, the notion of asymptotically even sequences is introduced. These conditions are satisfied if the Fourier coefficients $ \{ \hat f(n)\} $ are even $ (\hat f( - n) = \hat f(n))$, a case formally equivalent to cosine Fourier series. This paper applies the Tauberian method of Stanojević [3] separately to cosine and sine Fourier series and shows that the notion of asymptotic evenness can be circumvented in each case.


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

  • [1] A. N. Kolmogorov, Sur l'ordre de grandeur des coefficients de la série de Fourier-Lebesque, Bull. Internat. Acad. Polon. Sci. Lett. Cl. Sci. Math. Nat. Ser. A Sci. Math. (1923), 83-86.
  • [2] Č. V. Stanojević, Classes of $ {L^1}$-convergence of Fourier and Fourier-Stieljes series, Proc. Amer. Math. Soc. 82 (1981), 209-215. MR 609653 (82g:42006)
  • [3] -, Tauberian conditions for the $ {L^1}$-convergence of Fourier series, Trans. Amer. Math. Soc. 271 (1982), 237-244. MR 648089 (83f:42005)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1983-0691274-0
Keywords: $ {L^1}$-convergence of Fourier series
Article copyright: © Copyright 1983 American Mathematical Society

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