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

摘要

The general Poisson summation formula of harmonic analysis and analyticnumber theory can be regarded as a quadrature formula with remainder. Thepurpose of this investigation is to give estimates for this remainder based onthe classical modulus of smoothness and on an appropriate metric for describingthe distance of a function from a Bernstein space. Moreover, to be moreflexible when measuring the smoothness of a function, we consider Rieszderivatives of fractional order. It will be shown that the use of the abovemetric in connection with fractional order derivatives leads to estimates forthe remainder, which are best possible with respect to the order and theconstants.