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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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