## 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

### Keywords:

Lagrange interpolation
Full Text:

### References:

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