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\(q\)-parametric Bleimann Butzer and Hahn operators. (English) Zbl 1154.41012
Let the \(q\)-Bernstein polynomials are given by \[ B_{n,q}(f)(x) := \sum^n_{k=0}f \left({[k]\over[n]} \right) {n\choose k} \, x^k \prod^{n-k-1}_{s=0} (1-q^sx). \] In this paper, a \(q\)-analog of the Bleimann, Butzer and Hahn operators of the form \[ H_{n,q}(f)(x) := (\Phi^{-1}B_{n+1,q}\, \Phi)(f) (x) \] is proposed, where \( \Phi : C^*_p [0,\infty) \to C[0,1]\) is a suitable positive linear isomorphism. The authors study the properties of the \(q\)-BBH operators \(H_{n,q} \) and establish the rate of convergence. A Voronovskaja-type theorem and saturation of convergence for \(q\)-BBH operators for \( 0 < q < 1\) is discussed. Further, the convergence of the derivative of \(q\)-BBH operators is considered.

MSC:
41A36 Approximation by positive operators
41A40 Saturation in approximation theory
41A25 Rate of convergence, degree of approximation
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