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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

$ L\sp{1}$-convergence of Fourier series with coefficients of bounded variation


Authors: J. W. Garrett, C. S. Rees and Č. V. Stanojević
Journal: Proc. Amer. Math. Soc. 80 (1980), 423-430
MSC: Primary 42A28
MathSciNet review: 580997
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Abstract: It is shown that a Telyakovskii [2] theorem and its generalization are special cases of a corollary to a theorem of Garrett and Stanojević [4]. Another theorem of Garrett and Stanojević is generalized for the case of coefficients of bounded variation of order $ m \geqslant 1$.


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  • [1] A. N. Kolmogorov, Sur l'ordre de grandeur des coefficients de la série de Fourier-Lebesgue, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. (1923), 83-86.
  • [2] S. A. Telyakovskii, Concerning a sufficient condition of Sidon for the integrability of trigonometric series, Math. Notes 14 (1973), 742-748.
  • [3] S. Sidon, Hinreichende Bedingungen fur den Fourier-charakter einer trigonometrischen Reihe, J. London Math. Soc. 14 (1939), 158-160.
  • [4] J. W. Garrett and Č. V. Stanojević, Necessary and sufficient conditions for $ {L^1}$ convergence of trigonometric series, Proc. Amer. Math. Soc. 80 (1976), 68-71.
  • [5] Charles S. Rees and Caslav V. Stanojevic, Necessary and sufficient conditions for integrability of certain cosine sums, J. Math. Anal. Appl. 43 (1973), 579–586. MR 0322432 (48 #794)
  • [6] H. Mayer Marzuq, Necessary and sufficient conditions for integrability of certain cosine sums, Notices Amer. Math. Soc. 22 (1975), A-513.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1980-0580997-7
PII: S 0002-9939(1980)0580997-7
Keywords: $ {L^1}$-convergence of Fourier series
Article copyright: © Copyright 1980 American Mathematical Society