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Zbl 1236.41025
Ostrovska, Sofiya
Positive linear operators generated by analytic functions. (English)
[J] Proc. Indian Acad. Sci., Math. Sci. 117, No. 4, 485-493 (2007). ISSN 0253-4142; ISSN 0973-7685/e

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 $a_k x^k /\varphi(x)$, $k = 0, 1, \ldots$.\par The positive linear operator $(A_\varphi f)(x):= \bold E[f(\xi_x)]$ is studied. It is proved that if $\bold E(\xi_x) = x$, $\bold E(\xi_x ^{2}) = qx^{2} + bx + c,\quad q, b, c \in \bold R, q > 0$, 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 2000:: *41A36 Approximation by positive operators

Keywords: Szász-Mirakyan operator; positive operator; limit $q$-Bernstein operator; $q$-integers; Poisson distribution; totally positive sequence

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