Spectral method for derivingmultivariate Poisson summation formulae

Gusein Sh. Guseinov

doi:10.3934/cpaa.2013.12.359

Article Contents

2013, Volume 12, Issue 1: 359-373. Doi: 10.3934/cpaa.2013.12.359

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Spectral method for deriving multivariate Poisson summation formulae

Gusein Sh. Guseinov^1,

1.
Department of Mathematics, Atilim University, 06836 Incek, Ankara

Received: June 2011

Revised: March 2012

Published: January 2013

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Abstract

We show that using spectral theory of a finite family of pairwise commuting Laplace operators and the spectral properties of the periodic Laplace operator some analogues of the classical multivariate Poisson summation formula can be derived.

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Mathematics Subject Classification: Primary: 35J05, 35P05; Secondary: 35P10.

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References

[1]	T. M. Apostol, "Mathematical Analysis," 2nd edition, Addison-Wesley, 1974.
[2]	G. I. Arkhipov and V. N. Chubarikov, On some summation formulas, Vestnik Moskov. Univ. Ser. I Mat. Mekh., 5 (1987), 29-32 (Russian).
[3]	B. C. Berndt, Identities involving the coefficients of a class of Dirichlet series. V, Trans. Amer. Math. Soc., 160 (1971), 139-156.doi: 10.1090/S0002-9947-71-99991-0.
[4]	M. S. Birman and M. Z. Solomjak, "Spectral Theory of Self-Adjoint Operators in Hilbert Space," Dordrecht, 1987.
[5]	M. S. P. Eastham, "The Spectral Theory of Perodic Differential Equations," Scottish Academic Press, 1973.
[6]	G. H. Hardy, On the expression of a number as a sum of two squares, Quart. J. Math., 46 (1915), 263-283.
[7]	V. K. Ivanov, A generalization of the Voronoi-Hardy identity, Sibirsk Mat. Z., 3 (1962), 195-212 (Russian).
[8]	V. K. Ivanov, Higher-dimensional generalizations of the Euler summation formula, Izv. Vysš. Učebn. Zaved. Matematika, 6 (1963), 72-80 (Russian).
[9]	N. N. Lebedev, "Special Functions and Their Applications," Dover, 1972.
[10]	S. Leng, "Real Analysis," Addison-Wesley, 1969.
[11]	C. Müller, Eine Verallgemeinerung der Eulerschen Summenformel und ihre Anwendung auf Fragen der analytischen Zahlentheorie, Abh. Math. Sem. Univ. Hamburg, 19 (1954), 41-62.
[12]	C. Müller, Eine Formel der analytischen Zahlentheorie, Abh. Math. Sem. Univ. Hamburg, 19 (1954), 62-65.
[13]	C. Müller, Eine Erweiterung der Hardyschen Identität, Abh. Math. Sem. Univ. Hamburg, 19 (1954), 66-76.
[14]	M. N. Olevskii, On a summation formula connected with the Hankel transformation, Acad. Sci. USSR Doklady, 46 (1945), 387-391 (Russian).
[15]	E. C. Titchmarsh, "Eigenfunction Expansions Associated with Second-Order Differential Equations," Vol. 2, Oxford University Press, 1958.
[16]	G. F. Voronoi, Sur la dé veloppement, à l'aide des fonctions cylindriques, des sommes doubles $\sum f(pm^{2}+2qmn+rn^2)$, où $p m^2+2qmn+rn^2$ est une forme positive à coefficients entiers, (verhandlungen des Dritten Internat. Math.-Kongr, Heidelberg), Teubner, Leipzig, 1905, pp. 241-245.
[17]	V. V. Zhuk, Supplements to Poisson's summation formula and to the Hardy-Young theorem, Vestnik St. Petersburg Univ. Math., 25 (1992), 7-13.