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 ≥ 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.

[1]  Vladimir Temlyakov,et al.  CAMBRIDGE MONOGRAPHS ON APPLIED AND COMPUTATIONAL MATHEMATICS , 2022 .

[2]  J. Neumann.  On Rings of Operators. Reduction Theory , 1949 .

[3]  V. Müller,et al.  A product of three projections , 2014 .

[4]  P. Borodin.  Example of Divergence of a Greedy Algorithm with Respect to an Asymmetric Dictionary , 2021, Mathematical Notes.

[5]  C. Badea,et al.  The rate of convergence in the method of alternating projections , 2010, 1006.2047.

[6]  S. Reich,et al.  Unrestricted iterations of nonexpansive mappings in Hilbert space , 1992 .

[7]  E. Kopecká,et al.  Strange products of projections , 2015, 1508.05029.

[8]  J. Dye.  A generalization of a theorem of Amemiya and Ando on the convergence of random products of contractions in Hilbert space , 1989 .

[9]  Petr A. Borodin,et al.  Alternating projections, remotest projections, and greedy approximation , 2019, J. Approx. Theory.

[10]  E. Kopecká.  When products of projections diverge , 2019, Journal of the London Mathematical Society.

[11]  Stephen P. Boyd,et al.  Alternating Projections , 2022 .

[12]  P. Porcelli,et al.  On rings of operators , 1967 .

[13]  C. Badea,et al.  A generalization of the Friedrichs angle and the method of alternating projections , 2009 .

[14]  Greedy approximation by arbitrary sets , 2020 .

[15]  Heinz H. Bauschke,et al.  Characterizing arbitrarily slow convergence in the method of alternating projections , 2007, Int. Trans. Oper. Res..

[16]  Hein Hundal.  An alternating projection that does not converge in norm , 2004 .