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Proceedings of the American Mathematical Society

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

 

 

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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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