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A norm convergence result on random products of relaxed projections in Hilbert space

Author: H. H. Bauschke
Journal: Trans. Amer. Math. Soc. 347 (1995), 1365-1373
MSC: Primary 47H09; Secondary 46C99, 47N99, 92C55, 94A12
MathSciNet review: 1257097
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Abstract: Suppose $ X$ is a Hilbert space and $ {C_1}, \ldots ,{C_N}$ are closed convex intersecting subsets with projections $ {P_1}, \ldots ,{P_N}$. Suppose further $ r$ is a mapping from $ \mathbb{N}$ onto $ \{ 1, \ldots ,N\} $ that assumes every value infinitely often. We prove (a more general version of) the following result: If the $ N$-tuple $ ({C_1}, \ldots ,{C_N})$ is "innately boundedly regular", then the sequence $ ({x_n})$, defined by

$\displaystyle {x_0} \in X\;{\text{arbitrary,}}\quad {x_{n + 1}}: = {P_{r(n)}}{x_n},\quad {\text{for all}}\;n \geqslant 0,$

converges in norm to some point in $ \cap _{i = 1}^N{C_i}$.

Examples without the usual assumptions on compactness are given. Methods of this type have been used in areas like computerized tomography and signal processing.

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Additional Information

Keywords: Banach contraction, computerized tomography, convex feasibility problem, convex programming, convex set, Fejér monotone sequence, Hilbert space, image reconstruction, image recovery, innate bounded regularity, Kaczmarz's method, nonexpansive mapping, orthogonal projection, projection algorithm, projection method, projective mapping, random product, relaxation method, relaxed projection, signal processing, unrestricted iteration, unrestricted product
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