ABSTRACT
In this chapter, the author describe Gauss–Weierstrass averages of the multivariate Euler-type summation formula and first representations of Shannon-type theorems for regular regions without any geometric restriction, however, under smoothness assumptions originated by the ordinary Euler-type summation formula in Euclidean spaces. In author's approach, however, they are essentially inspired by C. Muller to transfer the theory of Fourier transforms to so-called spherically continuous functions. In doing so, they are led to structures and settings that are adequate for their pointwise efforts in lattice point theory involved with discontinuous functions relative to boundaries of regular regions. The authors introduce the bandlimited Gauss transform and Weierstrass transform as integrals over a regular region involving the kernels. By using Gauss–Weierstrass means, general geometries in Shannon–type sampling become discussible, however, under the assumption that the function under consideration is twice continuously differentiable. It should be remarked that the cardinal series shows an exponentially accelerated convergence.
