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Notes on special systems of orthogonal functions (IV): the orthogonal functions of whittaker's cardinal series

Published online by Cambridge University Press:  24 October 2008

G. H. Hardy
G. H. Hardy
Affiliation:
Trinity CollegeCambridge
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Extract

Suppose that n runs through all integral values, that (øn) is a system of normal orthogonal functions for the interval (−∞, ∞), and that ψn is the Fourier transform of øn. Then, by Parseval's theorem for Fourier transforms,

and (ψn) is also a normal orthogonal system.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 37 , Issue 4 , October 1941 , pp. 331 - 348
Copyright
Copyright © Cambridge Philosophical Society 1941

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References

REFERENCES

(1)Bateman, H.On the inversion of a definite integral. Proc. London Math. Soc. (2), 4 (1907), 461–98.CrossRefGoogle Scholar
(2)Hardy, G. H.An integral equation. Proc. London Math. Soc. (2), 7 (1909), 445–72.Google Scholar
(3)Hardy, G. H. and Titchmarsh, E. C.Solutions of some integral equations considered by Bateman, Kapteyn, Littlewood, and Milne. Proc. London Math. Soc. (2), 23 (1924), 1–26.Google Scholar
(4)Levinson, N.On non-harmonic Fourier series. Ann. Math. 37 (1936), 919–36.CrossRefGoogle Scholar
(5)Paley, R. E. A. C. and Wiener, N.Fourier transforms in the complex domain (New York, 1934).Google Scholar
(6)Plancherel, M. and Pólya, G.Fonctions entières et intégrales de Fourier multiples. Comm. math. Helvetici, 10 (1937), 110–63.CrossRefGoogle Scholar
(7)Pollard, S.The Stieltjes integral and its generalizations. Quart. J. Math. 49 (1923), 73–138.Google Scholar
(8)Titchmarsh, E. C.Introduction to the theory of Fourier integrals (Oxford, 1937).Google Scholar
(9)Whittaker, E. T.On the functions which are represented by the expansions of the interpolation theory. Proc. R.S. Edinb. 35 (1915), 181–194.Google Scholar
(10)Whittaker, J. M.Interpolatory function theory (Cambridge Tracts in Mathematics, no. 33, 1935).Google Scholar