Integrating \(n\)-unisolvent sets of functions
type: Item
instance of: scholarly article
title: Integrating \(n\)-unisolvent sets of functions
zbMATH Open document ID: 0918.41001
published in: Archiv der Mathematik
publication date: 1999-08-17T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 41A05
Mathematics Subject Classification ID: 30E05
Mathematics Subject Classification ID: 41A50
zbMATH DE Number: 1185645
zbMATH Keywords: completely \(n\)-valent functions
zbMATH Keywords: \(n\)-unisolvent sets of analytic functions
zbMATH Keywords: Chebyshev systems
author: Burkard Polster
reviewed by: T. Thrivikraman
MaRDI profile type: Publication
full work available at URL: s000130050201
OpenAlex ID: W2048436412
DOI: 10.1007/S000130050201
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description: scientific article; zbMATH DE number 1185645