On a Bernstein-Type Operator of Bleimann, Butzer and Hahn III

The operators $$ {L_n}\left( {f;x} \right) = \sum\limits_{k = 0}^n {{P_{n,k}}\left( x \right)f} \left( {\frac{k}{{n - k + 1}}} \right)\left( {x \geqslant 0,n \in N} \right) $$ with $${P_{n,k}}\left( x \right) = \left( {\begin{array}{*{20}{c}} n \\ k \end{array}} \right){x^k}{\left( {1 + x} \right)^{ - n}}$$ were introduced by Bleimann, Butzer and Hahn [3]. These operators are defined on C[0, ∞), the space of continuous functions on the unbounded interval [0, ∞) and have the property that they converge to f uniformly on any finite interval [a, b] contained in [0,∞) provided f is bounded and continuous on [0,∞). Using probabilistic arguments Khan [10] simplified and sharpened some of the results given in [3]. In [11] Kahn further showed that $${L_n}(f;x) \geqslant {L_{n + 1}}(f;x) \geqslant \ldots \geqslant f(x)$$ if f is convex; L n (f;x) is itself convex if f is a non-increasing convex function and $${L_n}(f;x) \in Li{p^\alpha }(A)(0 \leqslant \alpha \leqslant 1)$$ if and only if f ∈ Lip α(A) (0Ȧα≤1). The authors obtained in [8] the largest subclass of C[0, ∞) on which (Ln) defines a pointwise approximation process. The behaviour of the rational functions L n (f; z) for complex values of z outside [0, ∞) has also been determined in [8]. Totik [13] solved the saturation and so called non-optimal approximation problem for (L n ). A local saturation theorem for the operators (L n ) making use of the parabolic technique of De Vore [5] was obtained by the authors in [9]. A new Bernstein-type operator associated with the Polya distribution which includes as particular case the operator (L n ) is introduced by Adell et al. in [2] and the approximation properties of this operator concerning rate of convergence, preservation of Lipschitz constant and monotonic convergence under convexity are given.