  We introduce a new framework that yields spectral bounds on norms of
functions of transition maps for finite, homogeneous Markov chains. The
techniques employed work for bounded semigroups, in particular for classical as
well as for quantum Markov chains and they do not require additional
assumptions like detailed balance, irreducibility or aperiodicity. We use the
method in order to derive convergence bounds that improve significantly upon
known spectral bounds. The core technical observation is that power-boundedness
of transition maps of Markov chains enables a Wiener algebra functional
calculus in order to upper bound any norm of any holomorphic function of the
transition map. Finally, we discuss how general detailed balance conditions for
quantum Markov processes lead to spectral convergence bounds.
