David Seifert

Catalin Badea, David Seifert

Given $N\ge2$ closed subspaces $M_1,\dotsc, M_N$ of a Hilbert space $X$, let $P_k$ denote the orthogonal projection onto $M_k$, $1\le k\le N$. It is known that the sequence $(x_n)$, defined recursively by $x_0=x$ and $x_{n+1}=P_N\cdots P_1x_n$ for $n\ge0$, converges in norm to $P_Mx$ as $n\to\infty$ for all $x\in X$, where $P_M$ denotes the orthogonal projection onto $M=M_1\cap\dotsc\cap M_N$. Moreover, the rate of convergence is either exponentially fast for all $x\in X$ or as slow as one likes for appropriately chosen initial vectors $x\in X$. We give a new estimate in terms of natural geometric quantities on the rate of convergence in the case when it is known to be exponentially fast. More importantly, we then show that even when the rate of convergence is arbitrarily slow there exists, for each real number $α>0$, a dense subset $X_α$ of $X$ such that $\|x_n-P_Mx\|=o(n^{-α})$ as $n\to\infty$ for all $x\in X_α$. Furthermore, there exists another dense subset $X_\infty$ of $X$ such that, if $x\in X_\infty$, then $\|x_n-P_Mx\|=o(n^{-α})$ as $n\to\infty$ for all $α>0$. These latter results are obtained as consequences of general properties of Ritt operators. As a by-product, we also strengthen the unquantified convergence result by showing that $P_M x$ is in fact the limit of a series which converges unconditionally.

Catalin Badea, David Seifert

We present an extension of our earlier work [Ritt operators and convergence in the method of alternating projections, J. Approx. Theory, 205:133-148, 2016] by proving a general asymptotic result for orbits of an operator acting on a reflexive Banach space. This result is obtained under a condition involving the growth of the resolvent, and we also discuss conditions involving the location and the geometry of the numerical range of the operator. We then apply the general results to some classes of iterative projection methods in approximation theory, such as the Douglas-Rachford splitting method and, under suitable geometric conditions either on the ambient Banach space or on the projection operators, the method of alternating projections.

Oscar Darwin, Aashraya Jha, Souktik Roy et al.

The method of alternating projections involves projecting an element of a Hilbert space cyclically onto a collection of closed subspaces. It is known that the resulting sequence always converges in norm and that one can obtain estimates for the rate of convergence in terms of quantities describing the geometric relationship between the subspaces in question, namely their pairwise Friedrichs numbers. We consider the question of how best to order a given collection of subspaces so as to obtain the best estimate on the rate of convergence. We prove, by relating the ordering problem to a variant of the famous Travelling Salesman Problem, that correctness of a natural form of the Greedy Algorithm would imply that $\mathrm{P}=\mathrm{NP}$, before presenting a simple example which shows that, contrary to a claim made in the influential paper [Kayalar-Weinert, Math. Control Signals Systems, vol. 1(1), 1988], the result of the Greedy Algorithm is not in general optimal. We go on to establish sharp estimates on the degree to which the result of the Greedy Algorithm can differ from the optimal result. Underlying all of these results is a construction which shows that for any matrix whose entries satisfy certain natural assumptions it is possible to construct a Hilbert space and a collection of closed subspaces such that the pairwise Friedrichs numbers between the subspaces are given precisely by the entries of that matrix.