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Spectral method for deriving multivariate Poisson summation formulae
| 1. | Department of Mathematics, Atilim University, 06836 Incek, Ankara, Turkey |
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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