  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.
