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Lattice Functions in \(\mathbb{R}^q\)

  • Willi Freeden
  • Martin Gutting
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Part of the Applied and Numerical Harmonic Analysis book series (ANHA)

Abstract

If an attempt is made to generalize the one-dimensional theory to a higher dimensional case, we are confronted with pointwise convergence problems for the bilinear series of the multi-variate counterpart of G(Δ; ⋅). Nonetheless, as we have already seen in the one-dimensional case in Chap.9, we are able to circumvent any possible calamities by paying close attention to the defining constituents. However, the q-dimensional theory remains more complicated, since the characteristic singularity of the lattice function in lattice points becomes much harder to handle with increasing dimension. In conclusion, the proof of the Euler summation formula associated to the Laplace operator as well as the specification of sufficient criteria for validity of the Poisson summation formula is a matter of multi-dimensional potential theory. The results obtained in such a way are applicable in many branches, e.g., the calculation of certain lattice point sums involving charged particles, functional equations of Zeta and Theta functions, etc. Some of the applications are worked into exercises in Sect.10.9, where we also explain the interrelations between Green and spline functions. Our multi-periodic approach is based on the concepts of metaharmonic lattice point theory as presented in Freeden(2011).

Keywords

Lattice Point Zeta Function Laplace Operator Theta Function Summation Formula 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
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Authors and Affiliations

  • Willi Freeden
    • 1
  • Martin Gutting
    • 1
  1. 1.Geomathematics GroupUniversity of KaiserslauternKaiserslauternGermany

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