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Hilbert transforms and almost periodic functions

Published online by Cambridge University Press:  24 October 2008

J. Cossar
J. Cossar
Affiliation:
The UniversityEdinburgh
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Extract

The Hilbert transform, Hf, of a function f is defined by Hf = g, where

P denoting the Cauchy principal value and the integral being assumed to exist in some sense. When f is suitably restricted, Hf exists and

In the first part of Theorem 1 sufficient conditions are given for the validity of (1·2) rather more general than those of Wood ((6), p. 31). The present proof is based on the well-known condition of Riesz for the validity of (1·2), namely, that f is Lp(−∞, ∞) for some p > 1, and on the ‘Parseval’ relation (Lemma 3, (b)), which was used in a similar way by Hardy ((3), p. 110).

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 56 , Issue 4 , October 1960 , pp. 354 - 366
Copyright
Copyright © Cambridge Philosophical Society 1960

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References

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