Brushing up a theorem by Lehel Banjai on the convergence of Trapezoidal Rule Convolution Quadrature

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.
Comment: 32 pages, 1 figure; First part of the article will be submitted to Computers & Mathematics with Applications