Explicit representations for local Lagrangian numerical differentiation
Given the set \(T=\{t_0<t_1\cdots<t_n\}\), the corresponding Lagrangian interpolation formula for a function \(f\) is usually written as \(f(x)=\sum_{j=0}^n l_j(x)f(t_j)+R(x)\). Let \(\omega(x)=\prod_{i=0}^n(x-t_i)\). In the paper it is proved that the coefficients \(l_j^{(k)}(x)\) and \(\omega^{(k)}(x)\) can be represented explicitly by means of cycle indicator polynomials of a symmetric group. The authors present asymptotic expansions for remainders which refine the global estimate obtained by \textit{A. Shadrin} [J. Approximation Theory, 80, 25--49 (1995; Zbl 0814.41001)]. One of the local estimates, which is proved for \(f\in C^{n+1}(x)\) and \(\omega^{(k)}(x)\neq 0\), is the following (the case \(\omega^{(k)}(x)= 0\) is also considered) \[ R^{(k)}(x)=\omega^{(k)}(x)f^{(n+1)}(x)/(n+1)!+o(h^{n+1-k})\quad (h\to 0). \] In the last section the article presents icomplete formulas for numerical differentiation in an explicit form at the nodes and the non-nodes. These formulas hold when the nodes approach \(x\) in the sense that there exists a constant \(C>0\) independent of \(h\) such that \(T=T(h)\subset [x-Ch,x+Ch]\) (\(h\to 0\)).
scientific article; zbMATH DE number 5162314
scientific article