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Positive Bernstein-Sheffer operators. (English) Zbl 0835.41024
Summary: Let $$h(t)= \sum_{n\geq 1} h_n t^n$$, $$h_1>0$$, and $$\exp (xh (t))= \sum_{n\geq 0} p_n (x) t^n/ n!$$. For $$f\in C[0,1 ]$$, the associated Bernstein-Sheffer operator of degree $$n$$ is defined for $B^h_n f(x)= p_n^{-1} \sum_{k=0}^n f(k/ n) {n\choose k} p_k (x) p_{n-k} (1-x)$ where $$p_n= p_n (1)$$. We characterize functions $$h$$ for which $$B^h_n$$ is a positive operator for all $$n\geq 0$$. Then we give a necessary and sufficient condition insuring the uniform convergence of $$B^h_n f$$ to $$f$$. When $$h$$ is a polynomial, we give an upper bound for the error $$|f- B^h_n f|_\infty$$. We also discuss the behavior of $$B^h_n f$$ when $$h$$ is a series with a finite or infinite radius of convergence.

##### MSC:
 41A35 Approximation by operators (in particular, by integral operators)
##### Keywords:
Bernstein-Sheffer operator
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