## Image recovery by convex combinations of projections.(English)Zbl 0752.65045

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 $$T=T_ r\dots T_ 1$$.

### MSC:

 65J10 Numerical solutions to equations with linear operators 65Y05 Parallel numerical computation 46C05 Hilbert and pre-Hilbert spaces: geometry and topology (including spaces with semidefinite inner product) 47A50 Equations and inequalities involving linear operators, with vector unknowns
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### References:

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