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

Borwein, David

doi:10.1090/S0002-9939-1992-1062383-7

The nonlocal nature of the summability of Fourier series by certain absolute Riesz methods
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Proc. Amer. Math. Soc. 114 (1992), 89-94 Request permission

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.

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