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Summability of alternating gap series

  • J. P. Keating (a1) and J. B. Reade (a2)

Abstract

The Abel and Cesàro summabilities of two alternating gap series are investigated. We prove that the series is summable at x = 1 (in both senses), but that is not. In 1907, Hardy obtained essentially the same result for the latter series; our proof is shorter and more elementary: we use the Poisson summation formula to derive an explicit estimate for the size of the oscillations as x → 1_. This represents an example of a general method for determining the Abel summability of similar series.

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Copyright

References

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1.Abramowitz, M. and Stegun, I. A., Handbook of mathematical functions (Dover, New York, 1970).
2.Hardy, G. H., Q. J. Math. 38 (1907), 269288.
3.Hardy, G. H., Divergent series (Oxford University Press, 1949).
4.Katznelson, Y., An introduction to harmonic analysis (Wiley, Jerusalem, 1968).

Keywords

Summability of alternating gap series

  • J. P. Keating (a1) and J. B. Reade (a2)

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