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Abel Summability of Gap Series

Keating, J.P.; Reade, J.B.


Keyword(s): Poisson summation formula; gap series

Abstract: We show how the Poisson summation formula can be used to investigate the Abel summability of gap series. We use the method to prove that the series ?08 (-1)nxn2 is Abel summable, but that the series ?08 (-1)nx2n isn't. Hardy obtained essentially the same result as ours concerning the latter series in[1]. Our proof is shorter and more elementary. We give some numerical estimates for the size of the oscillation of the series ?08(-1)nxn2 as x? 1_.

6 Pages

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