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Notes on the Theory of Series (XVIII): on the Convergence of Fourier Series

Published online by Cambridge University Press:  24 October 2008

G. H. Hardy  and
J. E. Littlewood
G. H. Hardy
Affiliation:
Trinity College
J. E. Littlewood
Affiliation:
Trinity College
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Extract

1. In this note we give first our proof of a theorem (Theorem 1) which we stated in Note XIII. We then prove a new theorem (Theorem 2) which leads to another proof of the main theorem of Note XVII.

The first of these theorems requires some preliminary explanations. We are concerned with an integrable function f (θ) with the period 2π. We write

being the complex Fourier series of f (θ).

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 31 , Issue 3 , July 1935 , pp. 317 - 323
Copyright
Copyright © Cambridge Philosophical Society 1935

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References

REFERENCES

(1)Gabriel, R. M.: “The rearrangement of positive Fourier constants”, Proc. London Math. Soc. (2), 33 (1932), 32–51.CrossRefGoogle Scholar
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