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On the iterates of positive linear operators. (English) Zbl 1236.41002

Let \(U\) be a positive linear operator on \(C[0,1]\) that leaves the linear functions, \(P^1\), invariant. The paper presents a simple short proof of the following:
Theorem: If there is a continuous function \(f\) such that \(Uf-f\) has no zeros in \((1,0)\) then \(U^k g\) converges to the projection onto \(P^1\) that interpolates to \(g\) at \(0\) and \(1\).
The paper presents the theorem for more general domains for the operator than we stated above and applies to subspaces of bounded functions.

MSC:

41A05 Interpolation in approximation theory
41A36 Approximation by positive operators
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