The distance between the general Poisson summation formula and that for bandlimited functions; applications to quadrature formulae

TL;DR

This paper investigates the Poisson summation formula's remainder estimates for bandlimited functions, using smoothness measures and fractional derivatives to improve the bounds and applicability of quadrature formulas.

Contribution

It introduces a novel approach to estimate the Poisson summation formula's remainder using fractional derivatives and a new metric for function smoothness, enhancing previous bounds.

Findings

Derived optimal remainder estimates based on fractional derivatives.

Connected the metric of function distance to Bernstein spaces.

Provided bounds that are optimal with respect to order and constants.

Abstract

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.