Cesaro summation by spheres of lattice sums and Madelung constants. (English) Zbl 1476.11106
Summary: 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.
MSC:
| 11L03 | Trigonometric and exponential sums (general theory) |
| 42B08 | Summability in several variables |
| 35B10 | Periodic solutions to PDEs |
| 35R11 | Fractional partial differential equations |
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