The distance between the general Poisson summation formula and that for bandlimited functions; applications to quadrature formulae
Provides theoretically optimal error bounds for quadrature based on Poisson summation, relevant to numerical analysts and harmonic analysts.
The paper derives sharp estimates for the remainder in the Poisson summation formula when used as a quadrature rule, using fractional-order smoothness and a metric measuring distance from bandlimited functions, achieving optimal order and constants.
The general Poisson summation formula of harmonic analysis and analytic number theory can be regarded as a quadrature formula with remainder. The purpose of this investigation is to give estimates for this remainder based on the classical modulus of smoothness and on an appropriate metric for describing the distance of a function from a Bernstein space. Moreover, to be more flexible when measuring the smoothness of a function, we consider Riesz derivatives of fractional order. It will be shown that the use of the above metric in connection with fractional order derivatives leads to estimates for the remainder, which are best possible with respect to the order and the constants.