ABSTRACT
In this chapter, the authors intend for the use of Theta functions is to formulate multivariate Hardy–Landau-type lattice point formulas in Gauss–Weierstrass summability, thereby guaranteeing successfully the validity of both mere continuity for the weight functions and general geometry. The lattice point identities expressed in Gauss–Weierstrass summability serve as initial tools for developing bandlimited Shannon–type sampling formulas. Actually, the authors are able to guarantee that the resulting bandlimited, i.e., G-Fourier transformed values–based Shannon-type sampling formulas obtained by a “smoothing” integration process are also valid under non-summability assumptions. The approach enables to handle explicitly all situations resulting in over- and undersampling, where the aliasing error becomes concretely derivable within the multivariate lattice point context. The authors start with some information about multivariate Theta functions and functional equation and focus on Poisson-type summation over regular regions.
