Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-26T13:05:11.554Z Has data issue: false hasContentIssue false
  • English
  • Français

Hilbert transforms and almost periodic functions

Published online by Cambridge University Press:  24 October 2008

J. Cossar
J. Cossar
Affiliation:
The UniversityEdinburgh
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

(1)Besicovitch, A. S.Almost periodic functions (Cambridge, 1932).Google Scholar
(2)Bochner, S.Properties of Fourier series of almost periodic functions. Proc. Lond. Math. Soc. (2), 26 (1927), 433–52.CrossRefGoogle Scholar
(3)Hardy, G. H.On Hilbert transforms. Quart. J. Math., 3 (1932), 102–12.Google Scholar
(4)Riesz, M.Sur les fonctions conjuguées. Math. Z., 27 (1928), 218–44.CrossRefGoogle Scholar
(5)Titchmarsh, E. C.Introduction to the theory of Fourier integrals (Oxford, 1937).Google Scholar
(6)Wood, F. M.Reciprocal integral formulae. Proc. Lond. Math. Soc. (2), 29 (1929), 29–48.Google Scholar
(7)Zaanen, A. C.Linear analysis (Amsterdam, 1953, 1956).Google Scholar
(8)Zygmund, A.Trigonometrical series (Warsaw, 1935).Google Scholar
(9)Zygmundy, A.Trigonometric series, Vol. I (Cambridge, 1959).Google Scholar