Spectral method for deriving multivariate Poisson summation formulae

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January 2013, 12(1): 359-373. doi: 10.3934/cpaa.2013.12.359

Spectral method for deriving multivariate Poisson summation formulae

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

Received June 2011 Revised March 2012 Published September 2012

Abstract
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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.

Keywords: eigenfunction, eigenvalue, joint spectral resolution, Poisson summation formula., Commuting Laplace operators.

Mathematics Subject Classification: Primary: 35J05, 35P05; Secondary: 35P1.

Citation: Gusein Sh. Guseinov. Spectral method for deriving multivariate Poisson summation formulae. Communications on Pure & Applied Analysis, 2013, 12 (1) : 359-373. doi: 10.3934/cpaa.2013.12.359

References:

[1]	T. M. Apostol, "Mathematical Analysis,", 2nd edition, (1974). Google Scholar
[2]	G. I. Arkhipov and V. N. Chubarikov, On some summation formulas,, Vestnik Moskov. Univ. Ser. I Mat. Mekh., 5 (1987), 29. Google Scholar
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[5]	M. S. P. Eastham, "The Spectral Theory of Perodic Differential Equations,", Scottish Academic Press, (1973). Google Scholar
[6]	G. H. Hardy, On the expression of a number as a sum of two squares,, Quart. J. Math., 46 (1915), 263. Google Scholar
[7]	V. K. Ivanov, A generalization of the Voronoi-Hardy identity,, Sibirsk Mat. Z., 3 (1962), 195. Google Scholar
[8]	V. K. Ivanov, Higher-dimensional generalizations of the Euler summation formula,, Izv. Vys\vs. U\vcebn. Zaved. Matematika, 6 (1963), 72. Google Scholar
[9]	N. N. Lebedev, "Special Functions and Their Applications,", Dover, (1972). Google Scholar
[10]	S. Leng, "Real Analysis,", Addison-Wesley, (1969). Google Scholar
[11]	C. Müller, Eine Verallgemeinerung der Eulerschen Summenformel und ihre Anwendung auf Fragen der analytischen Zahlentheorie,, Abh. Math. Sem. Univ. Hamburg, 19 (1954), 41. Google Scholar
[12]	C. Müller, Eine Formel der analytischen Zahlentheorie,, Abh. Math. Sem. Univ. Hamburg, 19 (1954), 62. Google Scholar
[13]	C. Müller, Eine Erweiterung der Hardyschen Identität,, Abh. Math. Sem. Univ. Hamburg, 19 (1954), 66. Google Scholar
[14]	M. N. Olevskii, On a summation formula connected with the Hankel transformation,, Acad. Sci. USSR Doklady, 46 (1945), 387. Google Scholar
[15]	E. C. Titchmarsh, "Eigenfunction Expansions Associated with Second-Order Differential Equations,", Vol. 2, (1958). Google Scholar
[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, (1905), 241. Google Scholar
[17]	V. V. Zhuk, Supplements to Poisson's summation formula and to the Hardy-Young theorem,, Vestnik St. Petersburg Univ. Math., 25 (1992), 7. Google Scholar

show all references

References:

[1]	T. M. Apostol, "Mathematical Analysis,", 2nd edition, (1974). Google Scholar
[2]	G. I. Arkhipov and V. N. Chubarikov, On some summation formulas,, Vestnik Moskov. Univ. Ser. I Mat. Mekh., 5 (1987), 29. Google Scholar
[3]	B. C. Berndt, Identities involving the coefficients of a class of Dirichlet series. V,, Trans. Amer. Math. Soc., 160 (1971), 139. doi: 10.1090/S0002-9947-71-99991-0. Google Scholar
[4]	M. S. Birman and M. Z. Solomjak, "Spectral Theory of Self-Adjoint Operators in Hilbert Space,", Dordrecht, (1987). Google Scholar
[5]	M. S. P. Eastham, "The Spectral Theory of Perodic Differential Equations,", Scottish Academic Press, (1973). Google Scholar
[6]	G. H. Hardy, On the expression of a number as a sum of two squares,, Quart. J. Math., 46 (1915), 263. Google Scholar
[7]	V. K. Ivanov, A generalization of the Voronoi-Hardy identity,, Sibirsk Mat. Z., 3 (1962), 195. Google Scholar
[8]	V. K. Ivanov, Higher-dimensional generalizations of the Euler summation formula,, Izv. Vys\vs. U\vcebn. Zaved. Matematika, 6 (1963), 72. Google Scholar
[9]	N. N. Lebedev, "Special Functions and Their Applications,", Dover, (1972). Google Scholar
[10]	S. Leng, "Real Analysis,", Addison-Wesley, (1969). Google Scholar
[11]	C. Müller, Eine Verallgemeinerung der Eulerschen Summenformel und ihre Anwendung auf Fragen der analytischen Zahlentheorie,, Abh. Math. Sem. Univ. Hamburg, 19 (1954), 41. Google Scholar
[12]	C. Müller, Eine Formel der analytischen Zahlentheorie,, Abh. Math. Sem. Univ. Hamburg, 19 (1954), 62. Google Scholar
[13]	C. Müller, Eine Erweiterung der Hardyschen Identität,, Abh. Math. Sem. Univ. Hamburg, 19 (1954), 66. Google Scholar
[14]	M. N. Olevskii, On a summation formula connected with the Hankel transformation,, Acad. Sci. USSR Doklady, 46 (1945), 387. Google Scholar
[15]	E. C. Titchmarsh, "Eigenfunction Expansions Associated with Second-Order Differential Equations,", Vol. 2, (1958). Google Scholar
[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, (1905), 241. Google Scholar
[17]	V. V. Zhuk, Supplements to Poisson's summation formula and to the Hardy-Young theorem,, Vestnik St. Petersburg Univ. Math., 25 (1992), 7. Google Scholar

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