  Convolution quadrature (CQ) methods have enjoyed tremendous interest in
recent years as an efficient tool for solving time-domain wave problems in
unbounded domains via boundary integral equation techniques. In this paper we
consider CQ type formulations for the parallel space-time evaluation of
multistep or stiffly accurate Runge-Kutta rules for the wave equation. In
particular, we decouple the number of Laplace domain solves from the number of
time steps. This allows to overresolve in the Laplace domain by computing more
Laplace domain solutions solutions than there are time steps. We use techniques
from complex approximation theory to analyse the error of the CQ approximation
of the underlying time-stepping rule when overresolving in the Laplace domain
and show that the performance is intimately linked to the location of the poles
of the solution operator. Several examples using boundary integral equation
formulations in the Laplace domain are presented to illustrate the main
results.
