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On a theorem of Paley and Wiener

Published online by Cambridge University Press:  24 October 2008

G. H. Hardy
G. H. Hardy
Affiliation:
Trinity College
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Extract

The conjugate g of a periodic and integrable function f is not necessarily integrable, even when f is monotone inside its fundamental interval. Paley and Wiener, however, proved that g is integrable if f is monotone and odd. A simpler proof was given later by Zygmund‡.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 33 , Issue 1 , March 1937 , pp. 1 - 5
Copyright
Copyright © Cambridge Philosophical Society 1937

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References

* Suppose for example that the interval is (− π, π), and that

with π − a small.

† Trans. Amer. Math. Soc. 35 (1933), 348–56.Google Scholar

‡ Ibid. 36 (1934), 615–16.

§ Lebesgue integrable over (− ∞, ∞).

* For example,

* There is no special case of Theorem 1 corresponding to it exactly, since f (t) cannot be odd, increasing for all t, and integrable.

† This is not exactly a case of the Paley-Wiener theorem, but becomes one if we consider (0, 2π) instead of (− π, π).

‡ See Hardy, G. H. and Littlewood, J. E., Acta Math. 54 (1930), 99–102;CrossRefGoogle ScholarHardy, G. H., Littlewood, J. E. and Pólya, G., Inequalities, 169,Google Scholar Theorems 240, 241.

§ By theorems of Zygmund and M. Riesz. See Zygmund, , Trigonometrical Series, 150–151.Google Scholar