Abstract
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.
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Acknowledgments
The authors are grateful to the referee for useful comments.
Funding
The work of the first author is supported by the Russian Science Foundation under grant no. 22-21-00415, https://rscf.ru/project/22-21-00415/.
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Published in Russian in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2022, Vol. 319, pp. 64–72 https://doi.org/10.4213/tm4264.
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Borodin, P.A., Kopecká, E. Weak Limits of Consecutive Projections and of Greedy Steps. Proc. Steklov Inst. Math. 319, 56–63 (2022). https://doi.org/10.1134/S0081543822050054
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DOI: https://doi.org/10.1134/S0081543822050054