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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

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.
Citation: Gusein Sh. Guseinov. Spectral method for deriving multivariate Poisson summation formulae. Communications on Pure and 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.

[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.

show all references

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.

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