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On two theorems of S. Verblunsky

Published online by Cambridge University Press:  24 October 2008

Hubert Delange
Hubert Delange
Affiliation:
Faculté des SciencesUniversité de ClermontClermont-FerrandFrance
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Extract

In connexion with moment problems, S. Verblunsky proved the following two theorems:

Theorem I. (a) If f(x) is integrable in (−∞, +∞) and satisfies 0 ≤f(x) ≤ 1, then there exists a function σ(x), bounded and non-decreasing in (−∞, +∞), such that, for ζ = ξ + iη and η ≠ 0,

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 46 , Issue 1 , January 1950 , pp. 57 - 66
Copyright
Copyright © Cambridge Philosophical Society 1950

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References

* ‘Two moment problems for bounded functions’, Proc. Cambridge Phil. Soc. 42 (1946), 189–96.CrossRefGoogle Scholar

† ‘On the initial moments of a bounded function’, Proc. Cambridge Phil. Soc. 43 (1947), 275–9.CrossRefGoogle Scholar

* We omit the trivial case when f(x) = 0 almost everywhere.

† Notice that so that these inequalities define non-overlapping intervals.

* We omit the trivial case when σ(x) is constant.

* A very similar lemma was proved by Stieltjes, , Ann. Fac. Sci. Toulouse (1), 8 (1894), J72–J75.CrossRefGoogle Scholar

* A > 0 if a 1 > 0 and A = 0 if a 1 = 0.

* For instance, we may take