Approximation order of bivariate spline interpolation for arbitrary smoothness
type: Item
instance of: scholarly article
title: Approximation order of bivariate spline interpolation for arbitrary smoothness
zbMATH Open document ID: 0932.41002
DOI: 10.1016/S0377-0427(98)00004-1
author: O. Davydov
author: Günther Nürnberger
author: Frank Zeilfelder
published in: Journal of Computational and Applied Mathematics
publication date: 2000-03-21T00:00:00Z
review text: Consider the partition \(\Delta^1\) of the rectangle \(R= [a_0,b_0]\times [c_0,d_0]\) into triangles. This partition is formed by the subrectangles \(R_{i,j}= (x_{i-1}, x_i)\times (y_{i- 1},y_i)\), \(i= 1,\dots, n_1\), \(j= 1,\dots, n_2\), and the diagonals from the vertex \(z_{i- 1,j-1}= (x_{i-1}, y_{j-1})\) to \(z_{i,j}\). Let integers \(r\) and \(q\) with \(0\leq r< q\) be given and let \(\widetilde\Pi_q\) be the space of bivariate polynomials of total degree \(q\). The space \(S^r_q(\Delta^1)\) of bivariate splines of degree \(q\) and smoothness \(r\) is that of all functions \(s\in C^r(R)\) such that the restriction to each subtriangle of the partition \(\Delta^1\) is in \(\widetilde\Pi_q\). The paper under review deals with a Hermite interpolation problem where the space \(S^r_q(\Delta^1)\) of bivariate splines is involved. Theorem 1. Let \(T\) be an arbitrary subtriangle of \(\Delta^1\), and \(\widetilde T\) denote the adjacent subtriangle left of \(T\) in the same row if it exists and up of \(T\) otherwise. For each sufficiently differentiable function \(f\in C(R)\), there exists a unique solution \(s_f\in S^r_q(\Delta^1)\), \(q\geq 3.5r+ 1\), which satisfies the Hermite interpolation conditions described below. Condition A (Starting condition): \(D^\omega p(z)= D^\omega f(z)\), \(\omega= 0,\dots, q\), where \(z\) is a vertex of \(T\). Condition B: \(D^\omega p(z)= D^\omega f(z)\), \(\omega= 0,\dots, q-r-1\), where \(z\) is a vertex of \(T\) not belonging to \(\widetilde T\). Condition C: \(p_{x^\alpha y^\beta}(z)= f_{x^\alpha y^\beta}(z)\), \(\alpha\geq 0\), \(\beta\geq 0\), \(\alpha+ \beta\leq q-r-1\), \(\alpha+ 2\beta\leq 2q- 3r-2\), where \(z\) is the vertex of \(T\) not belonging to \(\widetilde T\). Condition D: \(D^\omega p(z)= D^\omega f(z)\), \(\omega= 0,\dots, q- 2r- 2\), where \(z\) is the midpoint of the diagonal of \(T\). Theorem 2. Let \(q,r\in\mathbb{N}\cup \{0\}\) be given such that \(q\geq 3.5r+1\). For each function \(f\in C^{q+ 1}(R)\), th…
Mathematics Subject Classification ID: 41A05
Mathematics Subject Classification ID: 41A15
Mathematics Subject Classification ID: 41A25
Mathematics Subject Classification ID: 41A63
zbMATH DE Number: 1326310
zbMATH Keywords: order of approximation
zbMATH Keywords: bivariate splines
zbMATH Keywords: Hermite interpolation
reviewed by: J. Illán
MaRDI profile type: Publication
cites work: Approximation power of smooth bivariate pp functions
cites work: A sharp upper bound on the approximation order of smooth bivariate pp functions
cites work: Q3831405
cites work: Stability of Optimal-Order Approximation by Bivariate Splines over Arbitrary Triangulations
cites work: Q4008655
cites work: Q4325863
cites work: Q3672421
cites work: On the approximation power of bivariate splines
cites work: Approximation order of bivariate spline interpolation
cites work: Lagrange and hermite interpolation by bivariate splines
cites work: Bivariate spline interpolation at grid points
cites work: Error analysis in interpolation by bivariate C1-splines
cites work: Q3737948
cites work: Q3728562
cites work: Interpolation de Lagrange par des splines quadratiques sur un quadrilatère de $\mathbb {R}^2$
description: scientific article; zbMATH DE number 1326310