Integrating \(n\)-unisolvent sets of functions
A set \(F\) of continuous functions from an interval \(I\) to \(\mathbb{R}\) is called \(n\)-unisolvent \((n\in\mathbb{N})\) if for any set of points \((x_i, y_i)\in I\times\mathbb{R}\), \(i=1,2,\dots,n\) with \(x_1<x_2<\cdots <x_n\), there exists exactly one \(f\in F\) that interpolates these points, i.e., \(f(x_i)=y_i\), \(i=1, \dots,n\). The set \(S(F)\) of all functions \(x\mapsto\int^x_af(t)dt+b\), \(a\in I\), \(b\in\mathbb{R}\), \(f\in F\) from \(I\to\mathbb{R}\) is the integral of \(F\). In this interesting paper the author proves that integrating an \(n\)-unisolvent set \(F\) on some open or semi-open interval gives an \((n+1)\)-unisolvent set \(S(F)\). Also if \(F\) solves the Hermite interpolation problem and the interval is open, then \(S(F)\) also solves the same problem. The author also attempts to generalise this to \(n\)-unisolvent sets of analytic functions defined on domains in the complex plane. The linear spans of Chebyshev systems associated with completely \((n-1)\)-valent functions are considered.
\(n\)-unisolvent sets of analytic functions
scientific article; zbMATH DE number 1185645
scientific article