ABSTRACT
This chapter demonstrates that the 1D Shannon sampling theory may be obtained by variants of (1D) Euler/Poisson summation formulas, namely lattice point identities resulting in so-called Hardy–Landau identities. It explains the classical Euler summation formula based on the Euler–Green function relative to the (1D) Laplace operator i.e. the second-order derivative, in its equivalence to the standard Poisson summation formula. The particular structure of the summation formula also captures the delicate details of the connection between integration, i.e., continuous summation, and its various discretizations. The Bernoulli function, i.e., in the jargon of mathematical physics, the Euler–Green function, acts as the connecting tool to convert a differential equation involving the 1D Laplace operator corresponding to periodic boundary conditions into an associated integral equation, i.e., the summation formula. The chapter recognizes the classical Shannon sampling theory in Euler/Poisson-type summation reflected approach.
