On Moving Averages

We show that the moving arithmetic average is closely connected to a Gauss-Seidel type fixed point method studied by H. H. Bauschke, X. Wang and C. J. S. Wylie [Fixed points of averages of resolvents: geometry and algorithms, SIAM J. Optimization 22 (2012) 24–40] and which was observed to converge only numerically. Our analysis establishes a rigorous proof of convergence of their algorithm in a special case; moreover, the limit is explicitly identified. Moving averages in Banach spaces and Kolmogorov means are also studied. Furthermore, we consider moving proximal averages and epi-averages of convex functions.