Alternating projections, remotest projections, and greedy approximation

[摘要] Let L-1, L-2, ..., L-K be a family of closed subspaces of a Hilbert space H, L-1 boolean AND ... boolean AND L-K = {0}; let P-k be the orthogonal projection onto L-k. We consider two types of consecutive projections of an element x(0) is an element of H: alternating projections T(n)x(0), where T = P-K circle ... circle P-1, and remotest projections x(n) defined recursively, x(n+1) being the remotest point for x(n) among P(1)x(n), ..., P(K)x(n). These x(n) can be interpreted as residuals in greedy approximation with respect to a special dictionary associated with L-1, L-2, ..., L-K. We establish parallels between convergence properties separately known for alternating projections, remotest projections, and greedy approximation in H. Here are some results. If L-1(perpendicular to) + ... + L-K(perpendicular to) = H, then x(n) -> 0 exponentially fast. In case L-1(perpendicular to) + ... + L-K(perpendicular to) not equal H, the convergence x(n) -> 0 can be arbitrarily slow for certain x(0). 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 vertical bar T(n)x(0)vertical bar <= C(x(0))n(-alpha(K)) for certain positive alpha(K) and all starting points x(0) is an element of L-1(perpendicular to)+ ... + L-K(perpendicular to). (C) 2020 Elsevier Inc. All rights reserved.

[发布日期] 2020-12-01

[关键词] Hilbert space;Products of projections;Greedy approximation;Rate of convergence