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Mercerian theorems

Published online by Cambridge University Press:  24 October 2008

H. R. Pitt
H. R. Pitt
Affiliation:
Peterhouse
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Extract

Suppose that k(x) is of bounded variation in (− ∞, ∞) and that s(x) is bounded and continuous in the same range. Let

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 34 , Issue 4 , October 1938 , pp. 510 - 520
Copyright
Copyright © Cambridge Philosophical Society 1938

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References

† Integrals with unspecified limits are over (− ∞, ∞).

† Pitt, H. R., “An extension of Wiener's general Tauberian theorem”, American Journal of Math. 60 (1938), 532–4CrossRefGoogle Scholar. We vise case (A) of the theorem.

‡ A singular function is continuous, of bounded variation and not constant, and has zero derivative almost everywhere.

† Wiener, N. and Pitt, H. R., “On absolutely convergent Fourier-Stieltjes transforms”, Duke Journal, 4 (1938), 420–36CrossRefGoogle Scholar, Theorem 1. When these Mercerian theorems were first proved I hoped that the class V* could be extended to include all functions of bounded variation. However, Theorem 3 of the above paper shows that some condition on the singular component of k(x) is essential, and it seems most natural to suppose that it vanishes altogether.

† See, for example, Bochner, S., Vorlesungen fiber Fouriersche Integrate (Leipzig, 1932)Google Scholar, Satz 18.

† The special case in which k(x) is absolutely continuous except at 0 has been treated by Paley, and Wiener, , “Fourier transforms in the complex domain”, American Math. Soc. Colloquium Publications, vol. 19 (New York, 1934)Google Scholar, Theorem XVII.

† Hausdorff, F., “Summationsmethoden und Momentfolgen”, Math. Zeitschrift, 37 (1921), 74–109.CrossRefGoogle Scholar

‡ Pitt, H. R., “General Tauberian theorems”, Proc. London Math. Soc. (2), 44 (1938), 243–88CrossRefGoogle Scholar, § 4·3.

§ See, for example, Bieberbach, L., Lehrbuch der Funktionentheorie (Springer, 1931), p. 138.Google Scholar

† Mercer, J., Proc. London Math. Soc. (2), 5 (1907), 206–24.CrossRefGoogle Scholar

† See Pitt, H. R., “General Tauberian theorems”, Proc. London Math. Soc. (2), 44 (1938), 243–88CrossRefGoogle Scholar, § 2. The particular form of Tauberian theorem for Borel summation which we use below is contained in Theorem 16 of the same paper.