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

Abstract

Suppose X X is a Hilbert space and C 1 , … , C N {C_1}, \ldots ,{C_N} are closed convex intersecting subsets with projections P 1 , … , P N {P_1}, \ldots ,{P_N} . Suppose further r r is a mapping from N \mathbb {N} onto { 1 , … , N } \{ 1, \ldots ,N\} that assumes every value infinitely often. We prove (a more general version of) the following result: If the N N -tuple ( C 1 , … , C N ) ({C_1}, \ldots ,{C_N}) is "innately boundedly regular", then the sequence ( x n ) ({x_n}) , defined by \[ x 0 ∈ X arbitrary, x n + 1 := P r ( n ) x n , for all n ⩾ 0 , {x_0} \in X\;{\text {arbitrary,}}\quad {x_{n + 1}}: = {P_{r(n)}}{x_n},\quad {\text {for all}}\;n \geqslant 0, \] converges in norm to some point in ∩ i = 1 N C i \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.