Box splines
type: Item
instance of: scholarly article
title: Box splines
zbMATH Open document ID: 0814.41012
author: Carl De Boor
author: Klaus Höllig
author: Sherman D. Riemenschneider
published in: Applied Mathematical Sciences
publication date: 1993-12-15T00:00:00Z
review text: From the authors' preface: Compactly supported smooth piecewise polynomial functions provide an efficient tool for the approximation of curves and surfaces and other smooth functions of one and several arguments. Since they are locally polynomial, they are easy to evaluate. Since they are smooth, they can be used when smoothness is required, as in the numerical solution of partial differential equations (in the finite element method) or the modeling of smooth surfaces (in computer aided geometric design). Since they are compactly supported, their linear span has the needed flexibility to approximate at all, and the system to be solved in the construction of approximations are `banded'. The construction of compactly supported smooth piecewise polynomials becomes ever more difficult as the dimension, \(s\), of their domain \(G\subseteq R^ s\), i.e. the number of arguments, increases. In the univariate case, there is only one kind of cell in any useful partition, namely, an interval, and its boundary consists of two separated points, across which polynomial pieces would have to be matched as one constructs a smooth piecewise polynomial function. This can be done easily, with the only limitation that the number of smoothness conditions across such a breakpoint should not exceed the polynomial degree (since thay would force the two joining polynomial pieces to coincide). In particular, on any partition, there are (nontrivial) compactly supported piecewise polynomials of degree \(\leq k\) and in \(C^{(k -1)}\), of which the univariate B-spline in the most useful example. However, when \(s>1\), then a useful partition may contain cells of various types (simplices and perturbations of parallelepipeds being the most common), and the boundary of a cell is not connected, but becomes an ever more important part of a cell as \(s\) increases (since its dimension differs from that of the cell itself only by 1). This makes the construction of low-degree compactly supported smooth p…
Mathematics Subject Classification ID: 41A15
zbMATH DE Number: 482831
zbMATH Keywords: B-spline
zbMATH Keywords: polyhedral
zbMATH Keywords: simplex splines
zbMATH Keywords: cardinal spline interpolation
zbMATH Keywords: multivariate Whittaker cardinal series
zbMATH Keywords: wavelets
zbMATH Keywords: discrete box splines
reviewed by: Elias Y. Deeba
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 482831