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On the rate of convergence of Bezier variant of Szász-Durrmeyer operators. (English) Zbl 1062.41024
Let $$p_{n,k}(x):=e^{-nx}(nx)^k/k!$$, and $$J_{n,k}(x):=\sum_{j=k}^\infty p_{n,j}(x)$$, $$n,k\geq0$$. The authors set $$Q^{(\alpha)}_{n,k}(x):=J^\alpha_{n,k}(x)-J^\alpha_{n,k+1}(x)$$, $$n,k\geq0$$, and introduce the following Durrmeyer variant of the Szász approximation operators on $$[0,\infty)$$, associated with a function $$f\in L[0,\infty)$$. Namely, $M_{n,\alpha}(f,x):=n\sum_{k=0}^\infty Q^{(\alpha)}_{n,k}(x)\int_0^\infty p_{n,k}(t)f(t)\,dt,\quad n=0,1,2,\dots.$ If $$\alpha=1$$, then the operators reduce to those defined by S. M. Mazhar and V. Totik [Acta Sci. Math. (Szeged), 49, 257–269 (1985; Zbl 0611.41013)]. Under the assumptions that $$f$$ is of bounded variation in every finite subinterval, and satisfies a growth condition $$| f(x)| \leq Ke^{\beta x}$$, the authors obtain pointwise estimates on how close $$M_{n,\alpha}(f,x)$$ is to the average $$(f(x+)+\alpha f(x-))/(\alpha+1)$$.

##### MSC:
 41A36 Approximation by positive operators 41A25 Rate of convergence, degree of approximation
##### Keywords:
Szász-Durrmeyer- type operators; rate of convergence
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##### References:
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