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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 n x) k=0 (a n x) k k!fk b n ,x[0,+),

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,+) functions satisfying lim x+ 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.


MSC:
41A36Approximation by positive operators
41A10Approximation by polynomials
41A63Multidimensional approximation problems
References:
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[2]Gadzhiev A D, Theorems of the type of P P Korovkin theorems,Math. Zametki 20(5) (1976) 781–786. English translation inMath. Notes 20(5–6) (1976) 996–998
[3]Gadzhiev A D, Positive linear operators in weighted spaces of functions of several variables,Izv. Akad. Nauk. SSR Ser. Fiz-Tekhn. Math. Nauk 4 (1980) 32–37
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