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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.

41A35 Approximation by operators (in particular, by integral operators)
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