A finite-difference approximation of the differentiation operator (Q2755749)

The author studies approximative properties of finite-difference operators of the most simple structure in Stechkin's problem of the best uniform approximation of the first oder differentiation operators on functional classes \(Q_n\) of functions \(f\in C({\mathbb R})\) with \(\|f^{(n)}\|\leqslant 1\), \(f^{(n-1)}\in AC({\mathbb R}_{\text{loc}})\), and compares them with the optimal (infinite-difference) operators. The following finite-difference operators are considered: \((Sf)(x)=(S_hf)(x)=\sum^m_{j=1}[A_j f(x+h_j)+A_{-j} f(x-h_j)]\), \(h_j=(2j-1)h\), \(h>0\), for odd~\(m\), and \((Sf)(x)=(S_hf)(x)=A_0 f(x)+\sum^m_{j=1}[A_j f(x+h_j)+A_{-j} f(x-h_j)]\), \(h_j=jh\), \(h>0\), for even~\(m\). It is shown that operators \(S\) with \(A_j=A_{-j}\) are optimal among all operators~\(S\). Let \(U_n(S)=\sup \{\|f'-Sf\|_C\mid f\in Q_n\}\), \(E(N,n)=\inf \{U_n(T)\mid \|T\|_{BL(C,C)}\leqslant N\}\). It is proved that (for \(h=1\)) \(\|S\|=\sum^{m-1}_{\nu =0} ({\Pi_\nu(2\nu+1)^2})^{-1}\), where \(\Pi_\nu= \frac {2^{2\nu}(\nu!)^2}{(2\nu+1)!}\) and \(\|S\|\to \pi/2\) as \(m\to +\infty\). Further, \(U_{2m+1}(S)=\frac {((2m-1)!!)^2}{(2m+1)!}\), \(\frac {(2m-1)!!}{(2m)!!(4m^2-1)} \leqslant U_{2m}(S)\leqslant \frac {((2m-1)!!)^2}{(2m!)(2m-1)}\). Let \(N>0\) and let \(h=h(N)>0\) be such that \(\|S_h\|=N\). Finally, approximative properties of the operators~\(S\) and the optimal infinite-difference operators are compared be means of \(U(S_{h(N)})/E(N,n)\), which is proved to be equal to \(U(S)/E(\|S\|,n)=:\sigma_n\), being independent on~\(N\). It is shown that \(\sigma_n =(\pi/2)^{n+\varepsilon_n}\), where \(\varepsilon_n=O(\sqrt n)\), \(n\to \infty\).