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A norm convergence result on random products of relaxed projections in Hilbert space
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by H. H. Bauschke PDF
Trans. Amer. Math. Soc. 347 (1995), 1365-1373 Request permission

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 \[ {x_0} \in X\;{\text {arbitrary,}}\quad {x_{n + 1}}: = {P_{r(n)}}{x_n},\quad {\text {for all}}\;n \geqslant 0,\] 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
  • © Copyright 1995 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 347 (1995), 1365-1373
  • MSC: Primary 47H09; Secondary 46C99, 47N99, 92C55, 94A12
  • DOI: https://doi.org/10.1090/S0002-9947-1995-1257097-1
  • MathSciNet review: 1257097