December  2021, 20(12): 4195-4208. doi: 10.3934/cpaa.2021153

Cesaro summation by spheres of lattice sums and Madelung constants

1. 

Department of Mathematics, University of Surrey, GU27XH, Guildford, UK

2. 

School of Mathematics and Statistics, Lanzhou University, Lanzhou, 730000, China

* Corresponding author

Received  July 2021 Published  December 2021 Early access  September 2021

Fund Project: The first author is supported by the LMS URB grant 1920-04. The second author is supported by the EPSRC grant EP/P024920/1

We study convergence of 3D lattice sums via expanding spheres. It is well-known that, in contrast to summation via expanding cubes, the expanding spheres method may lead to formally divergent series (this will be so e.g. for the classical NaCl-Madelung constant). In the present paper we prove that these series remain convergent in Cesaro sense. For the case of second order Cesaro summation, we present an elementary proof of convergence and the proof for first order Cesaro summation is more involved and is based on the Riemann localization for multi-dimensional Fourier series.

Citation: Benjamin Galbally, Sergey Zelik. Cesaro summation by spheres of lattice sums and Madelung constants. Communications on Pure and Applied Analysis, 2021, 20 (12) : 4195-4208. doi: 10.3934/cpaa.2021153
References:
[1]

N. Abatangelo and E. Valdinoci, Getting acquainted with the fractional Laplacian, in Contemporary Research in Elliptic PDEs and Related Topics, Springer, (2019), 1–105.

[2]

Sh. Alimov, R. Ashurov and A. Pulatov, Multiple Fourier Series and Fourier Integrals, Springer, (1992), 1–95. doi: 10.1007/978-3-662-06301-9_1.

[3]

M. Bartuccelli, J. Deane and S. Zelik, Asymptotic expansions and extremals for the critical Sobolev and Gagliardo-Nirenberg inequalities on a torus, Proc R. Soc. Edinb., 143 (2013), 445–482. doi: 10.1017/S0308210511000473.

[4] J. BorweinM. GlasserR. McPhedranJ. Wan and I. Zucker, Lattice Sums Then and Now, Cambridge, Cambridge University Press, 2013.  doi: 10.1017/CBO9781139626804.
[5]

A. Chaba and R. Pathria, Evaluation of lattice sums using Poisson's summation formula. II, J. Phys. A, 9 (1976), 1411–1423.

[6]

O. Emersleben, Über die Konvergenz der Reihen Epsteinscher Zetafunktionen, Math. Nachr., 4 (1950), 468–480. doi: 10.1002/mana.3210040140.

[7]

D. Gurarie, Symmetries and Laplacians, in: Introduction to Harmonic Analysis, Group Representations and Applications, 174, North-Holland, 1992.

[8] G. H. Hardy, Divergent Series, Oxford at the Clarendon Press, 1949. 
[9]

S. Marshall, A rapidly convergent modified Green's function for Laplace's equation in a rectangular region, Proc. R. Soc. Lond. A, 455 (1999), 1739–1766. doi: 10.1098/rspa.1999.0378.

[10]

S. Marshall, A periodic Green function for calculation of coloumbic lattice potentials, J. Phys. Condens. Matter, 12 (2000), 4575–4601. doi: 10.1088/0953-8984/12/21/304.

[11]

M. Ortiz Ramirez, Lattice points in d-dimensional spherical segments, Monatsh Math., 194 (2021), 167–179. doi: 10.1007/s00605-020-01447-y.

[12]

L. Roncal and P. Stinga, Transference of fractional Laplacian regularity, in Special Functions, Partial Differential Equations, and Harmonic Analysis, Springer, (2014), 203–212. doi: 10.1007/978-3-319-10545-1_14.

[13]

E. Stein, Localization and summability of multiple Fourier series, Acta Math., 100 (1958), 93–146. doi: 10.1007/BF02559603.

[14] G. Watson, A Treatise on the Theory of Bessel Functions, Cambridge University Press, Cambridge, England, 1995. 
[15]

S. Zelik and A. Ilyin, Asymptotics of the Green function and sharp interpolation inequalities, Uspekhi Mat. Nauk, 69 (2014), 23–76. doi: 10.1070/rm2014v069n02abeh004887.

show all references

References:
[1]

N. Abatangelo and E. Valdinoci, Getting acquainted with the fractional Laplacian, in Contemporary Research in Elliptic PDEs and Related Topics, Springer, (2019), 1–105.

[2]

Sh. Alimov, R. Ashurov and A. Pulatov, Multiple Fourier Series and Fourier Integrals, Springer, (1992), 1–95. doi: 10.1007/978-3-662-06301-9_1.

[3]

M. Bartuccelli, J. Deane and S. Zelik, Asymptotic expansions and extremals for the critical Sobolev and Gagliardo-Nirenberg inequalities on a torus, Proc R. Soc. Edinb., 143 (2013), 445–482. doi: 10.1017/S0308210511000473.

[4] J. BorweinM. GlasserR. McPhedranJ. Wan and I. Zucker, Lattice Sums Then and Now, Cambridge, Cambridge University Press, 2013.  doi: 10.1017/CBO9781139626804.
[5]

A. Chaba and R. Pathria, Evaluation of lattice sums using Poisson's summation formula. II, J. Phys. A, 9 (1976), 1411–1423.

[6]

O. Emersleben, Über die Konvergenz der Reihen Epsteinscher Zetafunktionen, Math. Nachr., 4 (1950), 468–480. doi: 10.1002/mana.3210040140.

[7]

D. Gurarie, Symmetries and Laplacians, in: Introduction to Harmonic Analysis, Group Representations and Applications, 174, North-Holland, 1992.

[8] G. H. Hardy, Divergent Series, Oxford at the Clarendon Press, 1949. 
[9]

S. Marshall, A rapidly convergent modified Green's function for Laplace's equation in a rectangular region, Proc. R. Soc. Lond. A, 455 (1999), 1739–1766. doi: 10.1098/rspa.1999.0378.

[10]

S. Marshall, A periodic Green function for calculation of coloumbic lattice potentials, J. Phys. Condens. Matter, 12 (2000), 4575–4601. doi: 10.1088/0953-8984/12/21/304.

[11]

M. Ortiz Ramirez, Lattice points in d-dimensional spherical segments, Monatsh Math., 194 (2021), 167–179. doi: 10.1007/s00605-020-01447-y.

[12]

L. Roncal and P. Stinga, Transference of fractional Laplacian regularity, in Special Functions, Partial Differential Equations, and Harmonic Analysis, Springer, (2014), 203–212. doi: 10.1007/978-3-319-10545-1_14.

[13]

E. Stein, Localization and summability of multiple Fourier series, Acta Math., 100 (1958), 93–146. doi: 10.1007/BF02559603.

[14] G. Watson, A Treatise on the Theory of Bessel Functions, Cambridge University Press, Cambridge, England, 1995. 
[15]

S. Zelik and A. Ilyin, Asymptotics of the Green function and sharp interpolation inequalities, Uspekhi Mat. Nauk, 69 (2014), 23–76. doi: 10.1070/rm2014v069n02abeh004887.

Figure 1.  A figure plotting $ N $th partial sums of (3.4) with $ a = 0 $ and $ s = \frac12 $ up to N = 5000
Figure 2.  A figure plotting $ N $th partial sums of (3.15) with $ a = 0 $ and $ s = \frac12 $ up to N = 5000
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