ABSTRACT

The essential obligations of the work are as follows: the authors first explain the lattice point approach to Shannon sampling for the univariate case. Even in the 1D context, number theoretically-motivated framework will be able to produce new sampling aspects, for example, in characterizing explicit over- and undersampling Shannon-type identities. The reason is that the multivariate counterparts of the 1D-Bernoulli polynomials, also called Euler–Green or lattice functions, constituting the essential ingredients of the Euler summation formula show a much stronger singularity behavior in higher dimensions than in the 1D-theory. In addition to the bandlimited theory of Shannon sampling, the approach based on Euler-type summation additionally attempts to come close to Shannon-type sampling of non-bandlimited functions. The essential ingredients of Shannon-type sampling results are based on a number of auxiliary means and tools, which in principle are well-known, but to some extent unusual and innovative in the context of Shannon-type sampling.