Israel J. Math. 194 (2013) 793-830.
Ian Melbourne and Dalia Terhesiu
Abstract 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 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 second (and higher) order asymptotics.
In the process, we
derive a higher order Tauberian theorem for scalar power series, which to our knowledge, has not previously been covered.