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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

Gusein Sh. Guseinov 1,

1. 

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

Received  June 2011 Revised  March 2012 Published  September 2012

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, Addison-Wesley, 1974.  Google Scholar

[2]

G. I. Arkhipov and V. N. Chubarikov, On some summation formulas, Vestnik Moskov. Univ. Ser. I Mat. Mekh., 5 (1987), 29-32 (Russian).  Google Scholar

[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.  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), 263-283. Google Scholar

[7]

V. K. Ivanov, A generalization of the Voronoi-Hardy identity, Sibirsk Mat. Z., 3 (1962), 195-212 (Russian).  Google Scholar

[8]

V. K. Ivanov, Higher-dimensional generalizations of the Euler summation formula, Izv. Vysš. Učebn. Zaved. Matematika, 6 (1963), 72-80 (Russian).  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), 41-62.  Google Scholar

[12]

C. Müller, Eine Formel der analytischen Zahlentheorie, Abh. Math. Sem. Univ. Hamburg, 19 (1954), 62-65.  Google Scholar

[13]

C. Müller, Eine Erweiterung der Hardyschen Identität, Abh. Math. Sem. Univ. Hamburg, 19 (1954), 66-76.  Google Scholar

[14]

M. N. Olevskii, On a summation formula connected with the Hankel transformation, Acad. Sci. USSR Doklady, 46 (1945), 387-391 (Russian). Google Scholar

[15]

E. C. Titchmarsh, "Eigenfunction Expansions Associated with Second-Order Differential Equations," Vol. 2, Oxford University Press, 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, Heidelberg), Teubner, Leipzig, 1905, pp. 241-245. 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), 7-13.  Google Scholar

show all references

References:
[1]

T. M. Apostol, "Mathematical Analysis," 2nd edition, Addison-Wesley, 1974.  Google Scholar

[2]

G. I. Arkhipov and V. N. Chubarikov, On some summation formulas, Vestnik Moskov. Univ. Ser. I Mat. Mekh., 5 (1987), 29-32 (Russian).  Google Scholar

[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.  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), 263-283. Google Scholar

[7]

V. K. Ivanov, A generalization of the Voronoi-Hardy identity, Sibirsk Mat. Z., 3 (1962), 195-212 (Russian).  Google Scholar

[8]

V. K. Ivanov, Higher-dimensional generalizations of the Euler summation formula, Izv. Vysš. Učebn. Zaved. Matematika, 6 (1963), 72-80 (Russian).  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), 41-62.  Google Scholar

[12]

C. Müller, Eine Formel der analytischen Zahlentheorie, Abh. Math. Sem. Univ. Hamburg, 19 (1954), 62-65.  Google Scholar

[13]

C. Müller, Eine Erweiterung der Hardyschen Identität, Abh. Math. Sem. Univ. Hamburg, 19 (1954), 66-76.  Google Scholar

[14]

M. N. Olevskii, On a summation formula connected with the Hankel transformation, Acad. Sci. USSR Doklady, 46 (1945), 387-391 (Russian). Google Scholar

[15]

E. C. Titchmarsh, "Eigenfunction Expansions Associated with Second-Order Differential Equations," Vol. 2, Oxford University Press, 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, Heidelberg), Teubner, Leipzig, 1905, pp. 241-245. 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), 7-13.  Google Scholar

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