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Absolute Riesz summability of Fourier series. I

Authors: G. D. Dikshit and C. S. Rees
Journal: Proc. Amer. Math. Soc. 82 (1981), 231-238
MSC: Primary 42A28; Secondary 40F05
MathSciNet review: 609657
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Abstract: In this paper we prove some theorems on the absolute summability of Fourier series which connect diverse $ \left\vert {C,\gamma } \right\vert$ results such as Bosanquet's classical theorem (1936), Mohanty (1952), and Ray (1970) and the recent $ \left\vert {R,\;\exp ({{(\log \omega )}^{\beta + 1}}),\gamma } \right\vert$ result of Nayak (1971).

It is also shown that in some sense some of the conclusions of the paper are the best possible.

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

  • [1] L. S. Bosanquet, Note on the absolute summability $ (C)$ of a Fourier series, J. London Math. Soc. 11 (1936), 11-15.
  • [2] L. S. Bosanquet and H. C. Chow, Some remarks on convergence and summability factors, J. London Math. Soc. 32 (1957), 73–82. MR 0083607
  • [3] K. Chandrasekharan and S. Minakshisundaram, Typical means, Oxford University Press, 1952. MR 0055458
  • [4] B. Kuttner, On the ‘second theorem of consistency’ for absolute Riesz summability, Proc. London Math. Soc. (3) 29 (1974), 17–32. MR 0352784
  • [5] R. Mohanty, Absolute Cesàro summability of a series associated with a Fourier series, Bull. Calcutta Math. Soc. 44 (1952), 152–154. MR 0056729
  • [6] M. K. Nayak, On the absolute summability and convergence of Fourier series and associated series, Proc. Cambridge Philos. Soc. 70 (1971), 421–433. MR 0290031
  • [7] B. K. Ray, On the absolute summability of some series related to a Fourier series, Proc. Cambridge Philos. Soc. 67 (1970), 29–45. MR 0412719

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Keywords: Fourier series, absolute Riesz summability, absolute Cesàro summability, absolute convergence, function of bounded variation, totally regular methods
Article copyright: © Copyright 1981 American Mathematical Society