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Positive linear operators generated by analytic functions. (English) Zbl 1236.41025
Summary: Let $$\varphi$$ be a power series with positive Taylor coefficients $$\{ak\}_k=0^{\infty}$$ and non-zero radius of convergence $$r \leq \infty$$. Let $$\xi_x, 0 \leq x < r$$ be a random variable whose values $$\alpha k$$ , $$k = 0, 1, \ldots$$, are independent of $$x$$ and taken with probabilities $$a_k x^k /\varphi(x)$$, $$k = 0, 1, \ldots$$.
The positive linear operator $$(A_\varphi f)(x):= \mathbf E[f(\xi_x)]$$ is studied. It is proved that if $$\mathbf E(\xi_x) = x$$, $$\mathbf E(\xi_x ^{2}) = qx^{2} + bx + c,\quad q, b, c \in \mathbf R, q > 0$$, then $$A_\varphi$$ reduces to the Szász-Mirakyan operator in the case $$q = 1$$, to the limit $$q$$-Bernstein operator in the case $$0 < q < 1$$, and to a modification of the Lupaş operator in the case $$q > 1$$.

##### MSC:
 41A36 Approximation by positive operators
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