Weak Limits of Consecutive Projections and of Greedy Steps

Petr A. Borodin ; Eva Kopecká

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Approximation Theory, Functional Analysis, and Applications, Tome 319 (2022), pp. 64-72.

Résumé

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 the 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 weak convergence in general remains open.

Détail

Keywords: projections, greedy approximations, convex set, dictionary, Hilbert space.

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