Open AccessCesaro summation by spheres of lattice sums and Madelung constants
Author(s) -
Benjamin Galbally,
Sergey Zelik
Publication year - 2021
Publication title -
communications on pure and applied analysis
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.077
H-Index - 42
eISSN - 1553-5258
pISSN - 1534-0392
DOI - 10.3934/cpaa.2021153
Subject(s) - spheres , madelung constant , mathematics , lattice (music) , fourier series , convergent series , pure mathematics , uniform convergence , euler summation , mathematical analysis , series (stratigraphy) , physics , crystal structure , lattice energy , computer science , power series , semi implicit euler method , computer network , paleontology , chemistry , backward euler method , bandwidth (computing) , astronomy , acoustics , discretization , biology , crystallography
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.