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On the rate of pointwise convergence of Szász-Mirakyan operators. (English) Zbl 0696.41020
Approximation and function spaces, Proc. 27th Semest., Warsaw/Pol. 1986, Banach Cent. Publ. 22, 323-330 (1989).
[For the entire collection see Zbl 0681.00013.]
Given arbitrary positive numbers M and $$\alpha$$, let $$A_ M^{\alpha}$$ denote the set of all complex-valued functions satisfying the growth condition $$| f(t)| \leq Mt^{\alpha t}$$ $$(0\leq t<\infty)$$. For $$f\in A_ M^{\alpha}$$ and $$x\in (0,\infty)$$, the Szász-Mirakyan operator is defined by $S_ u[f](x)=\sum^{\infty}_{k=0}f[k/n]p_ k(nx)\quad (n\in {\mathbb{N}}),$ where $$p_ k(t)=e^{-t} t^ k/k!.$$
Using the modulus of variation, the author presents some quantitative estimates of the rate of convergence of $$S_ n[f](x)$$ to $$(f(x+0)+f(x- 0)\}$$ for any function f in $$A_ M^{\alpha}$$, at every point $$x\in (0,\infty)$$ for which the one-sided limits of $$f(x\pm 0)$$ exist. For earlier work in this area, see Fuhua Cheng [J. Approximation Theory 40, 226-241 (1984; Zbl 0532.41026)].
Reviewer: H.R.Dowson
##### MSC:
 41A36 Approximation by positive operators 41A25 Rate of convergence, degree of approximation
##### Keywords:
Szász-Mirakyan operator