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On approximation of unbounded functions by linear combinations of modified Szász-Mirakian operators. (English) Zbl 0794.41014

The paper deals with the approximation of measurable functions \(f\) defined on \([0,\infty)\) with exponential growth and the property \(\int^ \infty_ 0e^{-nt}f(t)dt<\infty\) for \(n>n_ 0 (f)\) by \((k+1)\)-th order linear combinations (in the sense of C. P. May) of Szász-Mirakjan- Durrmeyer operators \(L_ n\). The main results are a Voronovskaja type asymptotic formula and a local direct estimate showing roughly speaking that the degree of approximation in the sup-norm on a compact subinterval \([a_ 1,b_ 1]\) can be at best \(O(n^{-k-1})\). The main tools for the proof of this last mentioned result are higher order Steklov means and classical moduli of smoothness on intervals \([a,b]\) with \(0<a<a_ 1<b_ 1<b<\infty\).

MSC:

41A36 Approximation by positive operators
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