Improved rate of approximation by modification of Baskakov operator. (English) Zbl 1524.41060
Summary: The optimal order of approximation, \(|L_n f(x)-f(x)|\) of a linear positive operator \(L_n f(x)\) is \(1/n\) and can not be improved however smooth the function may be. We remove the positivity of the Baskakov operator \(V_n(f;x)\) and introduce its three variants \(V_n^{M,i} (f;x)\), \(i=1,2,3\). We prove that the rates of approximation by these operators are improved from the linear order \(1/n\) to quadratic order \(1/n^2\) and then to cubic order \(1/n^3\) for sufficiently smooth functions.
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