ABSTRACT
This chapter offers the entry points for Shannon-type sampling over Euclidean spaces. Essential tools are the Poisson-type summation formulas in Euclidean spaces. The chapter investigates Shannon-type sampling for locally supported functions in more detail, and gives a study of the aliasing error. In analogy to the context implied by the Poisson-type summation formula for a regular region, the identities, respectively, offer two different entry points for Shannon-type sampling, i.e., functional values–based Shannon-type sampling and Fourier transformed values–based Shannon-type sampling. It should be noted that the role of functional values and Fourier transformed functional values of F is exchangeable, since the Fourier-transformed values of the Fourier transform canonically lead back to the functional values by virtue of the Fourier inversion formula. Nevertheless, a way is found in not-necessarily bandlimited sampling to detect the explicit aliasing error and to formulate Shannon-type approximate identities.
