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Zbl 1070.46013
Hundal, Hein S.
An alternating projection that does not converge in norm.
(English)
[J] Nonlinear Anal., Theory Methods Appl. 57, No. 1, A, 35-61 (2004). ISSN 0362-546X

Let $C_1$ and $C_2$ be two intersecting closed convex sets in a Hilbert space. Let $P_1$ and $P_2$ denote the corresponding projection operators. In 1933, von Neumann proved that the iterates produced by the sequence of alternating projections defined as ${y_n=(P_1 P_2)^n y_0}$ converge in norm to ${P_{C_1\cap C_2}(y_0)}$ when $C_1$ and $C_2$ are closed subspaces. {\it L. M.~Bregman} [Sov. Math., Dokl. 6, 688--692 (1965; Zbl 0142.16804)] showed that the iterates converge weakly to a point in ${C_1\cap C_2}$ for any pair of closed convex sets. In the paper under review, the author shows that alternating projections not always converge in the norm by constructing an explicit counterexample.
[Victor Milman (Minsk)]
MSC 2000:
*46C05 Geometry and topology of inner product spaces
41A65 Abstract approximation theory

Keywords: alternating projections; cyclic projections; feasible point; convex set

Citations: Zbl 0142.16804

Cited in: Zbl 1071.65082 Zbl 1065.47048

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