arXiv subjects

Eva Kopecka

Publications and source records attributed to Eva Kopecka.

3 recordsLinked to original sources

arXiv2021

Weak limits of consecutive projections and of greedy steps

Let $H$ be a Hilbert space. We investigate the properties of weak limit points of iterates of random projections onto $K\geq 2$ closed convex sets in $H$ and the parallel properties of weak limit points of residuals of random greedy approximation with respect to $K$ dictionaries. In the case of convex sets these properties imply weak convergence in all the cases known so far. In particular, we give a short proof of the theorem of Amemiya and Ando on weak convergence when the convex sets are subspaces. The question of the weak convergence in general remains open.

math.FA↗

arXiv2019

When products of projections diverge

Slow convergence of cyclic projections implies divergence of random projections and vice versa. Let $L_1,L_2,\dots,L_K$ be a family of $K$ closed subspaces of a Hilbert space. It is well known that although the cyclic product of the orthogonal projections on these spaces always converges in norm, random products might diverge. Moreover, in the cyclic case there is a dichotomy: the convergence is fast if and only if $L_1^{\perp}+\dots+L_K^{\perp}$ is closed; otherwise the convergence is arbitrarily slow. We prove a parallel to this result concerning random products: we characterize those families $L_1,\dots,L_K$ for which all random products converge using their geometric and combinatorial structure.

math.FA↗

arXiv2017

A note on alternating projections in Hilbert space

We provide a direct proof of a result regarding the asymptotic behavior of alternating nearest point projections onto two closed and convex sets in a Hilbert space. Our arguments are based on nonexpansive mapping theory.

math.FA↗