Hostname: page-component-586b7cd67f-r5fsc Total loading time: 0 Render date: 2024-11-26T07:34:58.045Z Has data issue: false hasContentIssue false

Extensions and applications of a tauberian theorem due to valiron

Published online by Cambridge University Press:  24 October 2008

M. E. Noble
Affiliation:
Queen's CollegeCambridge

Extract

1. Introduction. In the course of an important memoir on integral functions of finite order†, G. Valiron discusses ‘fonctions orientées’, that is, functions with zeros an such that arg an tends to a limit as n → ∞. He obtains results that include the following two theorems, in which Vρ(x) denotes a function of the form , where α1, …, αν are positive integers, and n(r) has its usual significance in the theory of integral functions:.

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 1951

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Ann. Fac. Sci. Toulouse (3), 5 (1914), 117.Google Scholar

Proc. London Math. Soc. (2), 26 (19261947), 185.Google Scholar

§ Quart. J. Math. 19 (1948), 90.Google Scholar

Ann. Math. 49 (1948), 200, 533.Google Scholar

C.R. Acad. Sci., Paris, 228 (1948), 1365.Google Scholar

Titchmarsh, E. C., Theory of functions (Oxford, 1932), p. 271.Google Scholar

Valiron, G., Integral functions (Toulouse, 1923), pp. 65–7.Google Scholar

Bull. Soc. Math. France, 35 (1907).Google Scholar

Proc. London Math. Soc. (2), 38 (1935), 178–9.Google Scholar