Paul L. Butzer

Carlo Bardaro, Paul L. Butzer, Ilaria Mantellini et al.

The general Poisson summation formula of Mellin analysis can be considered as a quadrature formula for the positive real axis with remainder. For Mellin bandlimited functions it becomes an exact quadrature formula. Our main aim is to study the speed of convergence to zero of the remainder for a function $f$ in terms of its distance from a space of Mellin bandlimited functions. The resulting estimates turn out to be of best possible order. Moreover, we characterize certain rates of convergence in terms of Mellin--Sobolev and Mellin--Hardy type spaces that contain $f$. Some numerical experiments illustrate and confirm these results.

Paul L. Butzer, Gerhard Schmeisser, Rudolf L. Stens

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.