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

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.