  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.
