• Positive Linear Operators Generated by Analytic Functions

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      https://www.ias.ac.in/article/fulltext/pmsc/117/04/0485-0493

    • Keywords

       

      Szász–Mirakyan operator; positive operator; limit 𝑞-Bernstein operator; 𝑞-integers; Poisson distribution; totally positive sequence.

    • Abstract

       

      Let 𝜑 be a power series with positive Taylor coefficients $\{a_k\}^\infty_{k=0}$ 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 $a_kx^k/\varphi(x), k=0,1,\ldots$

      The positive linear operator $(A_\varphi f)(x):=E[f(\xi_x)]$ is studied. It is proved that if $E(\xi_x)=x,E(\xi^2_x)=qx^2+bx+c,\, q, b, c\in R, q>0$, 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 $q>1$.

    • Author Affiliations

       

      Sofiya Ostrovska1

      1. Department of Mathematics, Atilim University, 06836 Incek, Ankara, Turkey
    • Dates

       
  • Proceedings – Mathematical Sciences | News

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