The multi-dimensional Von Neumann alternating direction search algorithm in C(B) and L1

Abstract

AbstractLet u1, …, um span a subspace U of a normed linear space X. Let ƒ ϵ X. An algorithm to find a best approximation to ƒ from U can be constructed by cyclically searching the one-dimensional spaces spanned by each ui. The best approximation from the one-dimensional spaces is subtracted from ƒ before searching in the next direction. This accumulating sum of best one-dimensional approximations is an algorithm for finding a best approximation from U. Variations of the method have been called Von Neumann alternating search, Diliberto-Strauss, and median polish algorithms. We first present the effects on the algorithm of smoothness and strict convexity of X. We then give detailed consideration to the algorithm in the two spaces C(B) and L1. The main results are characterizations for convergence to a best approximation