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Approximation by modified Szasz-Mirakjan operators on weighted spaces. (English) Zbl 1011.41012
The authors consider the modified Szász-Mirakyan operators ${S_n(f;x)}$ $$S_n(f;x):= \exp(-a_nx)\sum_{k=0}^{\infty}\frac{(a_n x)^k}{k!} f\left(\frac{k}{b_n}\right),\quad x\in [0,+\infty),$$ where $\{a_n\}$, $\{b_n\}$ are given increasing and unbounded sequences of positive numbers such that ${a_n/b_n=1+O(1/b_n)}$. Theorems of convergence of ${S_n(f;x)}$ to $f$ are obtained in spaces of continuous on ${[0,+\infty)}$ functions satisfying ${\lim_{x\to+\infty} f(x)/(1+x^2)=k(f)}$. The authors define a weighted modulus of continuity and by this modulus they obtain the rate of convergence of $S_n(f)$ to $f$. Korovkin-type theorems are widely used in the study. Similar results for functions of two variables are obtained.

41A36Approximation by positive operators
41A10Approximation by polynomials
41A63Multidimensional approximation problems
Full Text: DOI
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[4] Gadzhiev A D, Weighted approximation of continuous functions by linear operators on the whole real axis,Izv. Akad. Nauk. SSR Ser. Fiz-Tekhn. Math. Nauk 5 (1975) 41--45
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[8] Walczak Z, On certain modified Szasz--Mirakjan operators for functions of two variable,Demonstratio Math. XXXIII(1) (2000) 92--100 · Zbl 0960.41016