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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

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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


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:
  • MathSciNet review: 1257097