On the problem of numerical differentiation (Q909404)

A method is presented for approximation to k-th derivatives \(k<p\) of a p- times differentiable function f(x) on [-1,1] which has the property that it is stable with respect to small perturbations of f(x) with respect to the \(L_ 2[-1,1]\) norm. Let \(R_{\delta}f_{\delta}\) denote an approximation to \(f^{(k)}(x)\) constructed from any function \(f_{\delta}\) on a disk near f, \(\| f-f_{\delta}\| <\delta\). The authors prove that it is possible to choose \(c_ j\), N depending on \(f_{\delta}\) such that \(R_{\delta}f_{\delta}=\sum^{N}_{j=0}c_ jP_ j^{(k)}(x)\) satisfies \(| R_{\delta}f_{\delta}(x)- f^{(k)}(x)| \leq C(\epsilon)\delta^{1-1/2p-k/p}\), where \(- 1+\epsilon \leq x\leq 1-\epsilon\), \(\epsilon >0\) and \(P_ j(x)\) are the Legendre polynomials. Also \(\| R_{\delta}f_{\delta}-f^{(k)}\| \to 0\), \(\delta\) \(\to 0\). The result extends that of \textit{T. F. Dolgopolova}, \textit{V. K. Ivanov} for \(p=2\), \(k=1\) [Zh. Vychisl. Mat. Mat. Fiz. 6, 570-576 (1966; Zbl 0168.148)].