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Proof of a theorem of Paley

Published online by Cambridge University Press:  24 October 2008

A. Zygmund
A. Zygmund
Affiliation:
Wilno
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Extract

1. Let f(x) be a real function of period 2π, integrable L over (0, 2π), and let

By sn(x) and σn (x) we denote respectively the partial sums and the first arithmetic means of the series (1·1). Similarly, by and we denote the partial sums and the first arithmetic means of the series

conjugate to (1·1). By we mean the function conjugate to f(x), that is

where the integral is taken in the principal-value sense.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 34 , Issue 2 , April 1938 , pp. 125 - 133
Copyright
Copyright © Cambridge Philosophical Society 1938

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References

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