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A Remark on Fourier Transforms

Published online by Cambridge University Press:  24 October 2008

A. Zygmund
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1. Let f(x) be a complex function belonging to LP (−∞, ∞); i.e. let f(x) be measurable, and |f(x)|p integrable, over (−∞, ∞). The function

is called the Fourier transform of f(x), if the integral on the right exists, in some sense, for almost every value of y. It is well known that, if 1 ≤ p ≤ 2, the integral (1) converges in mean, with index p′ = p/(p – l)† i.e. that

where

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 32 , Issue 2 , May 1936 , pp. 321 - 327
Copyright
Copyright © Cambridge Philosophical Society 1936

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References

(1)Hardy, G. H. and Littlewood, J. E., “Some new properties of Fourier constants’, Journal London Math. Soc. 6 (1931), 3–9.CrossRefGoogle Scholar
(2)Kaezmerz, S. and Steinhaus, H.Theorie der Orthogonalreihen (Warsaw, 1935).Google Scholar
(3)Menchoff, D. “Sur les séries de fonctions orthogonales”, Fund. Math. 10 (1927), 375–420.CrossRefGoogle Scholar
(4)Paley, R. E. A. C., “Some theorems on orthogonal functions”, Studia Math. 3 (1931), 226–28.CrossRefGoogle Scholar
(5)Plancherel, M., “Contribution à l'étude de la représentation d'une fonction arbitraire par des intégrales définies”, Palermo Rendiconti, 30 (1910), 289–335.CrossRefGoogle Scholar
(6)Titchmarsh, E. C., “A contribution to the theory of Fourier transforms”, Proc. London Math. Soc. (2), 23 (1924), 279–89.Google Scholar
(7)Zygmund, A., Trigonometrical series (Warsaw, 1935).Google Scholar