Approximation with arbitrary order by Baskakov-type operators preserving exponential functions
Let us denote by \(C^{\ast }[0,\infty )\), the Banach space of all real-valued continuous functions on \([0,\infty )\) with the property that \( \lim_{x\rightarrow \infty }\) \(f(x)\) exists and is finite and let two sequences of positive real numbers \(\left( b_{n}\right) \) and \(\left( c_{n}\right) \) satisfy the conditions: \(0<c_{n}\leq \frac{n}{2},\) \( \lim_{n\rightarrow \infty }\frac{c_{n}}{n}=0\), \(\lim_{n\rightarrow \infty }b_{n}=+\infty .\) For \(a^{-x}\), with a fixed \(a>1\), the authors construct the following operator \[ L_{n}^{\left[ c_{n}\right] }(f,x)=\left( \frac{a^{\frac{c_{n}x}{n}}-a^{- \frac{1}{b_{n}}}}{1-a^{-\frac{1}{b_{n}}}}\right) ^{-\frac{n}{c_{n}} }\sum_{k=o}^{\infty }(-1)^{k}\left( \begin{array}{c} -\frac{n}{c_{n}} \\ k \end{array} \right) \left( \frac{a^{\frac{c_{n}x}{n}}-1}{a^{\frac{c_{n}x}{n}}-a^{-\frac{1 }{b_{n}}}}\right) ^{k}f\left( \frac{k}{b_{n}}\right) \] which preserves the constants and the function \(a^{-x}.\) As a measure of the deviation \(L_{n}^{\left[ c_{n}\right] }(f,x)-f\left( x\right) \) there is used the following modulus of continuity of \(f:\) \[ \omega ^{\ast }(f,\delta )=\sup_{ \begin{array}{c} t,x\geq 0 \\ |a^{-t}-a^{-x}|\leq \delta \end{array} }|f(t)-f(x)|. \] The main result proved in the paper forms \textbf{Theorem 1}. For every function \(f\in C^{\ast }[0,\infty )\), we have \[ \sup_{x\geq 0}\left\vert L_{n}^{\left[ c_{n}\right] }(f,x)-f\left( x\right) \right\vert \leq 2\omega ^{\ast }\left( f,\sqrt{M_{a}\max \left( \frac{1}{ b_{n}},\frac{c_{n}}{n}\right) }\right) ,{n> 1}, \] where \(M_{a}\) is a constant depending only on \(a.\) A Voronovskaya type result is formulated in \textbf{Theorem 2} and two corollaries are also proved.
scientific article; zbMATH DE number 7381216