Advanced Search

Query:

Fill in the form and click »Search«...

Format:

Display: entries per page entries

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

Comment on this Item PDF MathML XML ASCII DVI PS BibTeX Online Ordering Article Journal

	Scientific prize winners of the ICM 2010
	Overhang
	Lie groups, physics and geometry. An introduction for physicists, engineers and chemists.

Advanced Search

Zentralblatt MATH Berlin [Germany]