American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Global existence and nonexistence for the viscoelastic wave equation with nonlinear boundary damping-source interaction
  • CPAA Home
  • This Issue
  • Next Article
    Global existence and stability for a hydrodynamic system in the nematic liquid crystal flows
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, (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

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

[1]

Xiongping Dai, Yu Huang, Mingqing Xiao. Realization of joint spectral radius via Ergodic theory. Electronic Research Announcements, 2011, 18: 22-30. doi: 10.3934/era.2011.18.22
[2]

Sobhan Seyfaddini. Spectral killers and Poisson bracket invariants. Journal of Modern Dynamics, 2015, 9: 51-66. doi: 10.3934/jmd.2015.9.51
[3]

Alexei A. Ilyin. Lower bounds for the spectrum of the Laplace and Stokes operators. Discrete & Continuous Dynamical Systems - A, 2010, 28 (1) : 131-146. doi: 10.3934/dcds.2010.28.131
[4]

Victor Kozyakin. Iterative building of Barabanov norms and computation of the joint spectral radius for matrix sets. Discrete & Continuous Dynamical Systems - B, 2010, 14 (1) : 143-158. doi: 10.3934/dcdsb.2010.14.143
[5]

Daniel Guo, John Drake. A global semi-Lagrangian spectral model of shallow water equations with time-dependent variable resolution. Conference Publications, 2005, 2005 (Special) : 355-364. doi: 10.3934/proc.2005.2005.355
[6]

Isabeau Birindelli, Stefania Patrizi. A Neumann eigenvalue problem for fully nonlinear operators. Discrete & Continuous Dynamical Systems - A, 2010, 28 (2) : 845-863. doi: 10.3934/dcds.2010.28.845
[7]

Jean-Michel Rakotoson. Generalized eigenvalue problem for totally discontinuous operators. Discrete & Continuous Dynamical Systems - A, 2010, 28 (1) : 343-373. doi: 10.3934/dcds.2010.28.343
[8]

Marek Fila, Kazuhiro Ishige, Tatsuki Kawakami. Convergence to the Poisson kernel for the Laplace equation with a nonlinear dynamical boundary condition. Communications on Pure & Applied Analysis, 2012, 11 (3) : 1285-1301. doi: 10.3934/cpaa.2012.11.1285
[9]

Matthias Geissert, Horst Heck, Christof Trunk. $H^{\infty}$-calculus for a system of Laplace operators with mixed order boundary conditions. Discrete & Continuous Dynamical Systems - S, 2013, 6 (5) : 1259-1275. doi: 10.3934/dcdss.2013.6.1259
[10]

Sarah Constantin, Robert S. Strichartz, Miles Wheeler. Analysis of the Laplacian and spectral operators on the Vicsek set. Communications on Pure & Applied Analysis, 2011, 10 (1) : 1-44. doi: 10.3934/cpaa.2011.10.1
[11]

Damien Thomine. A spectral gap for transfer operators of piecewise expanding maps. Discrete & Continuous Dynamical Systems - A, 2011, 30 (3) : 917-944. doi: 10.3934/dcds.2011.30.917
[12]

Paul Loya and Jinsung Park. On gluing formulas for the spectral invariants of Dirac type operators. Electronic Research Announcements, 2005, 11: 1-11.

[13]

Kung-Ching Chang, Xuefeng Wang, Xie Wu. On the spectral theory of positive operators and PDE applications. Discrete & Continuous Dynamical Systems - A, 2019, 0 (0) : 0-0. doi: 10.3934/dcds.2020054
[14]

Victor Kozyakin. Minimax joint spectral radius and stabilizability of discrete-time linear switching control systems. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 3537-3556. doi: 10.3934/dcdsb.2018277
[15]

Isabeau Birindelli, Francoise Demengel. Eigenvalue, maximum principle and regularity for fully non linear homogeneous operators. Communications on Pure & Applied Analysis, 2007, 6 (2) : 335-366. doi: 10.3934/cpaa.2007.6.335
[16]

Robert T. Glassey, Walter A. Strauss. Perturbation of essential spectra of evolution operators and the Vlasov-Poisson-Boltzmann system. Discrete & Continuous Dynamical Systems - A, 1999, 5 (3) : 457-472. doi: 10.3934/dcds.1999.5.457
[17]

Oktay Veliev. Spectral expansion series with parenthesis for the nonself-adjoint periodic differential operators. Communications on Pure & Applied Analysis, 2019, 18 (1) : 397-424. doi: 10.3934/cpaa.2019020
[18]

Bertrand Lods, Mustapha Mokhtar-Kharroubi, Mohammed Sbihi. Spectral properties of general advection operators and weighted translation semigroups. Communications on Pure & Applied Analysis, 2009, 8 (5) : 1469-1492. doi: 10.3934/cpaa.2009.8.1469
[19]

Alexei Rybkin. On the boundary control approach to inverse spectral and scattering theory for Schrödinger operators. Inverse Problems & Imaging, 2009, 3 (1) : 139-149. doi: 10.3934/ipi.2009.3.139
[20]

David Damanik, Zheng Gan. Spectral properties of limit-periodic Schrödinger operators. Communications on Pure & Applied Analysis, 2011, 10 (3) : 859-871. doi: 10.3934/cpaa.2011.10.859

2018 Impact Factor: 0.925

Tools

Metrics

  • PDF downloads (9)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences