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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