Delay equations generate dynamical systems on infinite-dimensional state spaces. Their stability analysis is not immediate and reduction to finite dimension is often the only chance. Numerical collocation via pseudospectral techniques recently emerged as an efficient solution. In this part we analyze the application of these methods to discretize the evolution family associated to linear problems. The focus is on local stability of either equilibria and periodic orbits as well as on generic nonautonomous systems, for either delay differential and renewal equations.
