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The nonlocal nature of the summability of Fourier series by certain absolute Riesz methods


Author: David Borwein
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
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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 $ \vert {R,{\lambda _n},1} \vert$ is not a local property of the generating function. It is also proved, inter alia, that, for every $ \varepsilon > 0$, the $ \vert {R,{\lambda _n},1} \vert$ summability of the factored series $ \Sigma A_n ( t )\lambda _n^{ - \varepsilon }$ at any point is always a local property of the generating function.


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  • [1] S. N. Bhatt, An aspect of local property of $ \vert {R,\log n,1} \vert$ summability of factored Fourier series, Proc. Nat. Inst. Sci. India 26 (1960), 69-73. MR 0158218 (28:1444)
  • [2] H. Bor, Local property of $ {\vert {\overline N ,{p_n}} \vert _k}$ summability of factored Fourier series, Bull. Inst. Math. Acad. Sinica 17 (1989), 165-170. MR 1042427 (91e:42007)
  • [3] L. S. Bosanquet and H. Kestleman, The absolute convergence of series of integrals, Proc. London Math. Soc. (2) 45 (1939), 88-97.
  • [4] G. D. Dikshit, On the absolute Riesz summability factors of infinite series (I), Indian J. Math. 1 (1958), 33-40. MR 0104945 (21:3695)
  • [5] K. Matsumoto, Local property of summability $ \vert {R,{\lambda _n},1} \vert$, Tôhoku Math. J. (2) 8 (1956), 114-124. MR 0080200 (18:208b)
  • [6] K. N. Mishra, Multipliers for $ \vert {\overline N ,{p_n}} \vert$ summability of Fourier series, Bull. Inst. Math. Acad. Sinica 14 (1986), 431-438. MR 885371 (88e:42013)
  • [7] R. Mohanty, On the summability $ \vert {R,\log \omega ,1} \vert$ of a Fourier series, J. London Math. Soc. 25 (1950), 67-72. MR 0034466 (11:592b)
  • [8] -, On the absolute Riesz summability of Fourier series and allied series, Proc. London Math. Soc. 52 (1951), 295-321. MR 0041266 (12:822b)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1992-1062383-7
Keywords: Absolute summability, Riesz, weighted mean, Fourier series, local property
Article copyright: © Copyright 1992 American Mathematical Society

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