ABSTRACT
This chapter provides a different entry point to Shannon-type sampling over regular regions G. Firstly the cardinal series is constructed in such a way that it sums up functional values of the Fourier transform of the weight function over G in y-shifted lattice points, secondly the cardinal series sums up functional values of the weight function over G in a-shifted lattice points. The different entry points of the Poisson-type summation formula may be regarded as different points of departure to Shannon-type sampling. As a result, two different variants of Shannon-type sampling are obtained over regular regions in the sense of Gauss–Weierstrass summability, and later in ordinary sense. The chapter discusses some interesting examples (in Gauss–Weierstrass summability), which again demonstrate the close relationship of sampling and lattice point theory. The Fourier integral–based context provides a rich cornucopia of sampling capabilities.
