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On the Cesàro summability of Fourier series and Allied series

Published online by Cambridge University Press:  24 October 2008

R. E. A. C. Paley
R. E. A. C. Paley
Affiliation:
Trinity College
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Extract

For r>-1, let denote

If

we say that the series a0 + a1 + a2 +…+an+… is summable by Cesàro mean of order r, or more shortly summable (C, r) to sum s. If r >−1, and

we say that the series is summable by Rieszian mean of order r to the sum s. It has been shown that these two methods of summation are equivalent. Throughout this paper I shall deal with the Rieszian mean, but I shall retain the symbol (C, r). It is known† that if a series is summable (C, r), it is also summable (C, r′) to the same sum for all numbers r′ greater than r.

Type
Articles
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 26 , Issue 2 , April 1930 , pp. 173 - 203
Copyright
Copyright © Cambridge Philosophical Society 1930

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References

REFERENCES

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