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Series expansions and general transforms

Published online by Cambridge University Press:  24 October 2008

J. B. Miller
J. B. Miller
Affiliation:
University of New England, N.S.W.
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Extract

A number of writers have noticed Fourier sine or cosine transform relations between certain pairs of series. For example, Duffin(4) has shown that in a variety of circumstances

are a pair of sine transforms, and Guinand(7) that

are cosine transforms, using L2 theory. Other examples are given in (2) and (5); and in (6) power series and general transforms are considered.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 54 , Issue 3 , July 1958 , pp. 358 - 367
Copyright
Copyright © Cambridge Philosophical Society 1958

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References

REFERENCES

(1) Bateman Project, Tables of integral transforms vols. I and II.Google Scholar
(2)Boas, R. P.Sums representing Fourier transforms. Proc. Amer. Math. Soc. 3 (1952), 444–7.CrossRefGoogle Scholar
(3)Busbridge, I. W.A theory of general transforms of functions of the class LD (0, ∞) (1 < p ≤ 2). Quart. J. Math. (Oxford), 9 (1938), 148–60.CrossRefGoogle Scholar
(4)Duffin, R. J.Representation of Fourier integrals as sums, I, II, III. Bull. Amer. Math. Soc. 51 (1945), 447–55 and Proc. Amer. Math. Soc. 1 (1950), 250–5; 8 (1957), 272–7.CrossRefGoogle Scholar
(5)Goldberg, R. R. and Varga, R. S.Möbius inversion of Fourier transforms. Duke Math. J. 23 (1956), 553–9.CrossRefGoogle Scholar
(6)Goodspeed, F. M.Some generalisations of a formula of Ramanujan. Quart. J. Math. (Oxford), 10 (1939), 210–18.CrossRefGoogle Scholar
(7)Guinand, A. P.Some formulae for the Riemann zeta-function. J. Lond. Math. Soc. (1947), 14–18.CrossRefGoogle Scholar
(8)Hardy, G. H.On Dirichlet's divisor problem. Proc. Land. Math. Soc. 15 (1916), 1–25.Google Scholar
(9)Hardy, G. H. and Landau, E.The lattice points of a circle. Proc. Roy. Soc. A, 105 (1925), 244–58.Google Scholar
(10)Landau, E.Handbuch der Lehre von der Verteilung der Primzahlen (Leipzig, 1927).Google Scholar
(11)Miller, J. B.A symmetrical convergence theory for general transforms. Proc. Lond. Math. (in the Press).Google Scholar
(12)Oppenheim, A.Some identities in the theory of numbers. Proc. Lond. Math. Soc. (2), 26 (1927), 295–350.Google Scholar
(13)Titchmarsh, E. C.Theory of Fourier integrals (2nd ed.Oxford, 1948).Google Scholar
(14)Titchmarsh, E. C.Theory of the Riemann zeta function (Oxford 1951).Google Scholar
(15)Titchmarsh, E. C.Theory of functions (2nd ed.Oxford, 1947).Google Scholar
(16)Cramér, H.Mathematical methods of statistics (Princeton, 1946).Google Scholar