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doi: 10.3934/cpaa.2021153
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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 Revised  September 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 & Applied Analysis, 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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

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

[6]

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

[7]

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

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

[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.  Google Scholar

[11]

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

[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.  Google Scholar

[13]

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

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

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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

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

[6]

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

[7]

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

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

[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.  Google Scholar

[11]

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

[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.  Google Scholar

[13]

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

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

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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