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On the Closure of and Integral Functions

Published online by Cambridge University Press:  24 October 2008

Norman Levinson
Norman Levinson
Affiliation:
[Communicated by Miss M. L. Cartwright]
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Extract

1. A set of functions {øn (x)} is said to be closed L over an interval (a, b) if for an f (x) belonging to L

implies that f(x) = 0 almost everywhere. Here f(x) is a complex valued function of the real variable x.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 31 , Issue 3 , July 1935 , pp. 335 - 346
Copyright
Copyright © Cambridge Philosophical Society 1935

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References

* Wiener, and Paley, , “On two problems of Pólya”, Trans. American Math. Soc. 35 (1933), 782–3.Google Scholar

† Pólya, , “Untersuchungen über Lücken und Singularitäten von Potenzreihen”, Math. Zeitschrift, 29 (1929), 549–640.CrossRefGoogle Scholar

* Ostrowski, A., Acta Lit. ac Sc. Reg. Univ. Hung. Franc. Jos. 1 (1923), 80–87.Google Scholar

† See Titchmarsh, E. C., Proc. London Math. Soc. (2), 25 (1926), 283–302.CrossRefGoogle Scholar

* Wiener, and Paley, , “On entire functions”, Trans. American Math. Soc. 35 (1933), Theorem I, p. 769Google Scholar, and Wiener, and Paley, , “Fourier transforms in the complex domain”, Amer. Math. Soc. Coll. Pub. 19 (1934), Theorem XXI.Google Scholar

† Cartwright, M. C., Proc. London Math. Soc. (2), 38 (1935), 179.Google Scholar

‡ Cartwright, M. C., Proc. Cambridge Phil. Soc. 34 (1935), 347–350.CrossRefGoogle Scholar

§ Titchmarsh, , Theory of functions, p. 125.Google Scholar

∥ Loc. cit. p. 130.

Loc. cit. p. 177.

* In case f (z) has zeros inside the contour equation (3.00) is modified by the addition of logarithmic terms on the left which can be found by handling (3.02) just as in the proof of Carleman's theorem. These terms can be shown to be positive, and in this case (3.00) becomes.

* That follows from an application of the theorem of Fatou on sequences of positive integrals.