Image recovery by convex combinations of projections (Q1814151)

The functional analytic question discussed in this paper is: For which \(T\) one has weak convergence of the sequences \(\{T^ nx\}^ \infty_{n=0}\) to a common fixed point of a finite number of projections \(P_ 1,\dots,P_ r\) (onto convex closed subsets \(C_ 1,\dots,C_ r\)) in a Hilbert space. It is shown via more abstract results that one may choose \(T=\alpha_ 0id+\sum^ r_{i=1}\alpha_ iT_ i\) with \(T_ i=id+\lambda_ i(P_ i-id)\), \(0<\lambda_ i<2\), \(\alpha_ j>0\), \(\sum^ r_ 0\alpha_ j=1\). It is argued that this choice is more suitable for parallel computer implementation than the classical \(T=T_ r\dots T_ 1\).