  In this paper, we establish sublinear and linear convergence of fixed point
iterations generated by averaged operators in a Hilbert space. Our results are
achieved under a bounded H\"older regularity assumption which generalizes the
well-known notion of bounded linear regularity. As an application of our
results, we provide a convergence rate analysis for Krasnoselskii-Mann
iterations, the cyclic projection algorithm, and the Douglas-Rachford
feasibility algorithm along with some variants. In the important case in which
the underlying sets are convex sets described by convex polynomials in a finite
dimensional space, we show that the H\"older regularity properties are
automatically satisfied, from which sublinear convergence follows.
