Alternating projections, remotest projections, and greedy approximation

Abstract

Let L1,L2,…,LK be a family of closed subspaces of a Hilbert space H, L1∩⋯∩LK={0}; let Pk be the orthogonal projection onto Lk. We consider two types of consecutive projections of an element x0∈H: alternating projections Tnx0, where T=PK∘⋯∘P1, and remotest projections xn defined recursively, xn+1 being the remotest point for xn among P1xn,…,PKxn. These xn can be interpreted as residuals in greedy approximation with respect to a special dictionary associated with L1,L2,…,LK. We establish parallels between convergence properties separately known for alternating projections, remotest projections, and greedy approximation in H. Here are some results. If L1⊥+⋯+LK⊥=H, then xn→0 exponentially fast. In case L1⊥+⋯+LK⊥≠H, the convergence xn→0 can be arbitrarily slow for certain x0. Such a dichotomy, exponential rate of convergence everywhere on H, or arbitrarily slow convergence for certain starting elements, is valid for greedy approximation with respect to general dictionaries. The dichotomy was known for alternating projections. Using the methods developed for greedy approximation we prove that |Tnx0|≤C(x0)n−α(K) for certain positive α(K) and all starting points x0∈L1⊥+⋯+LK⊥.