Bivariate Hermite interpolation and numerical curves
type: Item
instance of: scholarly article
title: Bivariate Hermite interpolation and numerical curves
zbMATH Open document ID: 0857.41001
author: H. V. Gevorgian
author: Hakop Hakopian
author: Artur Sahakian
published in: Journal of Approximation Theory
publication date: 1996-08-05T00:00:00Z
review text: Here the authors return to their favorite theme of Hermite interpolation by bivariate algebraic polynomials of degree \(n\) given by a scheme \({\mathcal N}=\{m_1,\dots,m_s;n\}\) where \(m_1,\dots,m_s\) are non-negative integers, \(n\) is the degree of the polynomial and \(s\) is the length of \(\mathcal N\). The set of all schemes is denoted by \(\mathcal J\). The interpolation parameters are the values of a function and its partial derivative up to some order \(n_\nu-1\) at the nodes \(z_\nu=(x_\nu,y_\nu)\), \(\nu=1,\dots,s\), where \(n_\nu\) is the multiplicity of \(z_\nu\). A scheme \({\mathcal N}\in {\mathcal J}\) is called an interpolation scheme if \(\sum^s_{\nu=1} \overline n_\nu=\overline{(n+1)}\), where \(\overline n_\nu= {n_\nu(n_\nu+1)\over 2}\). A scheme \(\mathcal N\) is called regular if \(({\mathcal N},{\mathcal L})\) is correct for at least one node set \(\mathcal L\). It is called singular if it is not correct for any node set. Geometrically the singularity of \(\mathcal N\) means that for any node set \(\mathcal L\) there exists an algebraic curve of degree \(\leq n\) passing through \(\mathcal L\) with multiplicity \(m_\nu\) \((\nu=1,\dots,s)\) at \(z_\nu\). The authors introduce the idea of quadratic transformation and reduction of schemes. A quadratic transformation \(\mathcal N\) into \(N^*\) while reduction takes \(N\) into \(N^*\). The significance of quadratic transformation and reduction is illustrated by the following theorem which was proved in 1992 [\textit{A. A. Akopyan}, \textit{O. V. Gevorkyan}, and \textit{A. A. Saakyan}, Russ. Acad. Sci., Sb., Math. 76, No. 1, 211-223 (1993; Zbl 0782.41001)]: Theorem. Let \({\mathcal N}=\{m_1,\dots,m_s; n\}\in S\). Then (i) if \(n_1+n_2=n+r\geq n+1\), \(n_1\leq n\), \(n_2\leq n\), then the schemes \(N\) and \(N^*\) are both singular or both regular, (ii) if \(n_i+n_j\leq n\), \(1\leq i< j\leq3\), then the schemes \(N\) and \(N^*\) are both regular or both singular. The intersection (product) of two…
Mathematics Subject Classification ID: 41A05
Mathematics Subject Classification ID: 41A63
Mathematics Subject Classification ID: 41A10
zbMATH DE Number: 894210
zbMATH Keywords: Hermite interpolation by bivariate algebraic polynomials
reviewed by: A. Sharma
MaRDI profile type: Publication
full work available at URL: jath.1996.0044
OpenAlex ID: W2059798770
DOI: 10.1006/JATH.1996.0044
description: scientific article; zbMATH DE number 894210