Box Splines

• Carl de Boor
• Klaus Höllig
• Sherman Riemenschneider
Textbook

Part of the Applied Mathematical Sciences book series (AMS, volume 98)

1. Front Matter
Pages i-xvii
2. Carl de Boor, Klaus Höllig, Sherman Riemenschneider
Pages 1-31
3. Carl de Boor, Klaus Höllig, Sherman Riemenschneider
Pages 33-60
4. Carl de Boor, Klaus Höllig, Sherman Riemenschneider
Pages 61-78
5. Carl de Boor, Klaus Höllig, Sherman Riemenschneider
Pages 79-103
6. Carl de Boor, Klaus Höllig, Sherman Riemenschneider
Pages 105-136
7. Carl de Boor, Klaus Höllig, Sherman Riemenschneider
Pages 137-158
8. Carl de Boor, Klaus Höllig, Sherman Riemenschneider
Pages 159-174
9. Back Matter
Pages 175-201

Introduction

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 sur­ faces (in Computer Aided Geometric Design). Since they are compactly supported, their linear span has the needed flexibility to approximate at all, and the systems 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 ~ IRs, 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 num­ ber of smoothness conditions across such a breakpoint should not exceed the polynomial degree (since that would force the two joining polynomial pieces to coincide). In particular, on any partition, there are (nontrivial) compactly supported piecewise polynomials of degree ~ k and in C(k-l), of which the univariate B-spline is the most useful example.

Keywords

Division algebra algorithms equation numerical analysis proof

Authors and affiliations

• Carl de Boor
• 1
• Klaus Höllig
• 2
• Sherman Riemenschneider
• 3
2. 2.Math Institut A der UniversitätStuttgart 80Germany
3. 3.Department of MathematicsUniversity of AlbertaEdmontonCanada

Bibliographic information

• DOI https://doi.org/10.1007/978-1-4757-2244-4
• Copyright Information Springer-Verlag New York 1993
• Publisher Name Springer, New York, NY
• eBook Packages
• Print ISBN 978-1-4419-2834-4
• Online ISBN 978-1-4757-2244-4
• Series Print ISSN 0066-5452
• Buy this book on publisher's site