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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2024 MCQ for Proceedings of the American Mathematical Society is 0.85.

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The nonlocal nature of the summability of Fourier series by certain absolute Riesz methods
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by David Borwein
Proc. Amer. Math. Soc. 114 (1992), 89-94
DOI: https://doi.org/10.1090/S0002-9939-1992-1062383-7
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Abstract:

It is proved that for a large class of sequences $\{ {{\lambda _n}} \}$ the summability at a point of a Fourier series $\Sigma A_n ( t )$ by the absolute Riesz method $| {R,{\lambda _n},1} |$ is not a local property of the generating function. It is also proved, inter alia, that, for every $\varepsilon > 0$, the $| {R,{\lambda _n},1} |$ summability of the factored series $\Sigma A_n ( t )\lambda _n^{ - \varepsilon }$ at any point is always a local property of the generating function.
References
  • Shri Nivas Bhatt, An aspect of local property of $| R,\,\textrm {log}\,n, 1|$ summability of the factored Fourier series, Proc. Nat. Inst. Sci. India Part A 26 (1960), 69–73. MR 158218
  • HΓΌseyin Bor, Local property of $|\overline N,p_n|_k$ summability of factored Fourier series, Bull. Inst. Math. Acad. Sinica 17 (1989), no.Β 2, 165–170. MR 1042427
    • L. S. Bosanquet and H. Kestleman, The absolute convergence of series of integrals, Proc. London Math. Soc. (2) 45 (1939), 88-97.
  • G. D. Dikshit, On the absolute Riesz summability factors of infinite series. I, Indian J. Math. 1 (1958), no.Β 1, 33–40 (1958). MR 104945
  • Kishi Matsumoto, Local property of the summability $|R,\lambda _n,1|$, Tohoku Math. J. (2) 8 (1956), 114–124. MR 80200, DOI 10.2748/tmj/1178245014
  • K. N. Mishra, Multipliers for $|\overline N,p_n|$ summability of Fourier series, Bull. Inst. Math. Acad. Sinica 14 (1986), no.Β 4, 431–438. MR 885371
  • R. Mohanty, On the summability $| R,\textrm {log}, w,1|$ of a Fourier series, J. London Math. Soc. 25 (1950), 67–72. MR 34466, DOI 10.1112/jlms/s1-25.1.67
  • R. Mohanty, On the absolute Riesz summability of Fourier series and allied series, Proc. London Math. Soc. (2) 52 (1951), 295–320. MR 41266, DOI 10.1112/plms/s2-52.4.295
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Bibliographic Information
  • © Copyright 1992 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 114 (1992), 89-94
  • MSC: Primary 42A28
  • DOI: https://doi.org/10.1090/S0002-9939-1992-1062383-7
  • MathSciNet review: 1062383