# A Norm Convergence Result on Random Products of Relaxed Projections in Hilbert Space

Bauschke, Heinz H. (1994) A Norm Convergence Result on Random Products of Relaxed Projections in Hilbert Space. [Preprint]

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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 ${\Bbb N}$ onto $\{1,\ldots,N\}$ that assumes every value infinitely often. We prove (a more general version of) the following result: \begin{quote} If the $N$-tuple $(C_1,\ldots,C_N)$ is innately boundedly regular'', then the sequence $(x_n)$, defined by $$x_0 \in X ~\mbox{arbitrary}, ~~~ x_{n+1} := P_{r(n)}x_n, ~~\mbox{for all n \geq 0},$$ converges in norm to some point in $\bigcap_{i=1}^{N} C_i$. \end{quote} Examples without the usual assumptions on compactness are given. Methods of this type have been used in areas like computerized tomography and signal processing.