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The consistency of cardinal series

Published online by Cambridge University Press:  24 October 2008

M. E. Noble
M. E. Noble
Affiliation:
The University Nottingham
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Extract

1. Whittaker (7), generalizing a result of Ferrar(3), showed that the cardinal series based on the positive and negative integers is consistent in the sense that, if

and an integral function f (x) is denned by

then provided 0 < λ > 1

In this note I show that results of Paley- Wiener, Levinson and others on biorthogonal series can be made to yield a consistency theorem for cardinal series based on sequences λn, where

that is, series

where

We use Fourier transform technique and need hypotheses, a little more restrictive than Whittaker's, which would reduce in the case .

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 50 , Issue 1 , January 1954 , pp. 139 - 142
Copyright
Copyright © Cambridge Philosophical Society 1954

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References

REFERENCES

(1)Duffin, R. J. and Eachus, J. J.Some notes on an expansion theorem of Paley and Wiener. Bull. Amer. math. Soc. 48 (1942), 850–5.Google Scholar
(2)Duffin, R. J. and Schaeffer, A. C.A class of nonhannonic Fourier series. Trans. Amer. math. Soc. 72 (1952), 341–66.CrossRefGoogle Scholar
(3)Ferrar, W. L.On the consistency of the cardinal function of interpolation. Proc. roy. Soc. Edinb. 47 (1927), 230–42.CrossRefGoogle Scholar
(4)Hilding, S. H.Linear methods in the theory of complete sets in a Hilbert space. Ark. Mat. Antar. Fys. 35A (1948), no. 38.Google Scholar
(5)Levinson, N.Gap and density theorems. American Math. Colloquium, xxvi (New York, 1940).Google Scholar
(6)Paley, R. E. A. C. and Wiener, N.Fourier transforms in the complex domain. American Math. Colloquium, XIX (New York, 1934).Google Scholar
(7)Whittaker, J. M.Interpolatory function theory (Camb. Tracts Math., 1935).Google Scholar