  This paper deals with the approximation of the spectrum of linear and
nonautonomous delay differential equations through the reduction of the
relevant evolution semigroup from infinite to finite dimension. The focus is
placed on classic collocation, even though the requirements that a numerical
scheme has to fulfill in order to allow for a correct approximation of the
spectral elements are recalled. This choice, motivated by the analyticity of
the underlying eigenfunctions, allows for a convergence of infinite order, as
rigorously demonstrated through a priori error bounds when Chebyshev nodes are
adopted. Fundamental applications such as determination of asymptotic stability
of equilibria (autonomous case) and limit cycles (periodic case) follow at
once.
