A recurrence relation for generalized divided differences with respect to ECT-systems
Consider an extended complete Chebyshev system (ECT-system) \(\{ u_{0},u_{1},\dots ,u_{n}\} \) defined on an interval \(G\subset\mathbb{R}\) or an a region \(G\subset\mathbb{C}\). One considers the Hermite type interpolation problem: given the nodes \(a_{0},a_{1},\dots ,a_{n}\in G\) and a function \(f\) on \(G\) find \(p\in \operatorname {span}\{ u_{0},u_{1},\dots ,u_{n}\} \) such that \(p^{\mu_{i}( a_{i}) }( a_{i}) =f^{\mu _{i}( a_{i}) }( a_{i}) ,\) \(i=0,1,\dots ,n,\) where \(\mu_{i}( a) \) is the multiplicity of a in \(( a_{0} ,a_{1},\dots ,a_{i-1}) \). This problem has a unique solution which can be written in Newton form. The coefficients of \(p\) written in Newton form are called generalized divided differences. One proves a recurrence formula for these generalized divided differences and one shows that they are continuous functions of the nodes. These general results are exemplified on the case of the ECT-system given by \(u_{j}( z) :=\Phi( z) e^{j\delta z},\) \(j=0,1,\dots ,n,\) \(\delta\in\mathbb{C}\), \(\delta\neq 0\), fixed and \(\Phi\in\mathbb{C}^{n}( G,\mathbb{C}) \), \(\Phi( z) \neq 0\) on \(G\subset\mathbb{C},\) and for a function \(f:G\to\mathbb{C}\) which is sufficiently smooth on the nodes \(\alpha_{0},\alpha_{1} ,\dots ,\alpha_{n}\in G.\)
scientific article; zbMATH DE number 1421572