ABSTRACT
After the well-known material involving algebraic polynomials and splines, presented here only for recapitulation and motivation, we are interested in explaining different types of integration involving periodic polynomials in one-dimensional Euclidean space ℝ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315195674/823ed023-dd6c-41bb-a456-22174b64a780/content/imath3_1.tif"/> . We start with the classical Euler summation formula for the operator of the second derivative, i.e., the Laplace operator in one dimension and “periodic boundary conditions”. Later on, we extend the Euler summation formula to Helmholtz operators. Both variants will be used to develop sufficient conditions for the validity of the Poisson summation formula in one dimension. As a matter of fact, Euler and Poisson summation formulas turn out to be equivalent in one dimension (see the introduction of W. Freeden [2011]). However, in the context of approximate integration, they take a different role. Remainder terms in Euler summation formulas include derivatives of the integrand under consideration, whereas error terms in Poisson summation formulas are free of derivatives.
