Cardinal Hermite interpolation using positive definite functions
type: Item
instance of: scholarly article
title: Cardinal Hermite interpolation using positive definite functions
zbMATH Open document ID: 0810.41005
DOI: 10.1007/BF02140686
published in: Numerical Algorithms
publication date: 1995-04-18T00:00:00Z
review text: Cardinal Hermite interpolation problem (CHIP) was first studied by \textit{P. R. Lipow} and \textit{I. J. Schoenberg} [Linear Algebra Appl. 6, 273-304 (1973; Zbl 0246.41015)]. This paper is a continuation of this study. The main result is the following: Denote by \(S_{2m-1,r}\) the set of all functions that are polynomials of degree \(2m-1\) in each of the intervals \([j, j+1]\) \((j\in\mathbb{Z})\). Then for any sequence of data \(d_ 1= (d_{1,j}),\dots, d_ r= (d_{r,j})\), \(j\in \mathbb{Z}\), of polynomial growth there is a unique \(s_{2m-1} (x)\in S_{2m-1}\) such that \(s_{2m-1}^{(k)} (j)= d_{k,j}\) \((k=0,1,\dots, r-1)\) for all integers \(j\). This paper is a continuation of the study of CHIP. It contains necessary and sufficient conditions for solvability of CHIP for \(\ell^{(p)}\)-data \((p=1,\infty)\) and further study of the ``solvability'' condition. Several examples are presented here and the relation of results of the paper to other papers in this field is explained.
Mathematics Subject Classification ID: 41A05
zbMATH DE Number: 640745
zbMATH Keywords: Hermite interpolation
author: Xingping Sun
reviewed by: Tibor Šalát
MaRDI profile type: Publication
cites work: Q5558293
cites work: Fundamental Solutions for Multivariate Difference Equations
cites work: Scattered data interpolation by linear combinations of translates of conditionally positive definite functions
cites work: Q4122353
cites work: Fourier Transforms of B-Splines and Fundamental Splines for Cardinal Hermite Interpolations
cites work: Cardinal interpolation and spline functions. III: Cardinal Hermite interpolation
cites work: Polyharmonic cardinal splines
cites work: Polyharmonic cardinal splines: A minimization property
cites work: Cardinal Hermite Spline Interpolation: Convergence as the Degree Tends to Infinity
cites work: Cardinal Hermite interpolation with box splines
cites work: Cardinal Hermite interpolation with box splines. II
cites work: Q5640160
cites work: On the Gaps between the Values of a Rational Quadratic Polynomial
full work available at URL: https://doi.org/10.1007/bf02140686
OpenAlex ID: W2057114685
description: scientific article; zbMATH DE number 640745