  This document is made up of two different units. One of them is a regular
terse research article, whereas the other one is the detailed and independently
written explanations for the paper, so that readers of the short paper do not
need to go over all the cumbersome computations. The goal is to clarify the
dependence with respect to the time variable of some estimates about the
convergence of the Trapezoidal Rule based Convolution Quadrature method applied
to hyperbolic problems. This requires a careful investigation of the article of
Lehel Banjai where the first convergence estimates were introduced, and of some
technical results from a classical paper of Christian Lubich.
