  We generalize the proof of Karamata's Theorem by the method of approximation
by polynomials to the operator case. As a consequence, we offer a simple proof
of \emph{uniform dual ergodicity} for a very large class of dynamical systems
with infinite measure, and we obtain bounds on the convergence rate.
  In many cases of interest, including the Pomeau-Manneville family of
intermittency maps, the estimates obtained through real Tauberian remainder
theory are very weak. Building on the techniques of complex Tauberian remainder
theory, we develop a method that provides \emph{second (and higher) order
asymptotics}. In the process, we derive a \emph{higher order Tauberian theorem}
for scalar power series, which to our knowledge, has not previously been
covered.
