# Multivariate Birkhoff Interpolation

• Authors
• Rudolph A. Lorentz
Book

Part of the Lecture Notes in Mathematics book series (LNM, volume 1516)

1. Front Matter
Pages I-IX
2. Rudolph A. Lorentz
Pages 1-3
3. Rudolph A. Lorentz
Pages 4-8
4. Rudolph A. Lorentz
Pages 9-22
5. Rudolph A. Lorentz
Pages 23-49
6. Rudolph A. Lorentz
Pages 50-61
7. Rudolph A. Lorentz
Pages 62-71
8. Rudolph A. Lorentz
Pages 72-74
9. Rudolph A. Lorentz
Pages 75-89
10. Rudolph A. Lorentz
Pages 90-102
11. Rudolph A. Lorentz
Pages 103-138
12. Rudolph A. Lorentz
Pages 139-155
13. Rudolph A. Lorentz
Pages 156-161
14. Rudolph A. Lorentz
Pages 162-170
15. Back Matter
Pages 171-192

### Introduction

The subject of this book is Lagrange, Hermite and Birkhoff (lacunary Hermite) interpolation by multivariate algebraic polynomials. It unifies and extends a new algorithmic approach to this subject which was introduced and developed by G.G. Lorentz and the author. One particularly interesting feature of this algorithmic approach is that it obviates the necessity of finding a formula for the Vandermonde determinant of a multivariate interpolation in order to determine its regularity (which formulas are practically unknown anyways) by determining the regularity through simple geometric manipulations in the Euclidean space. Although interpolation is a classical problem, it is surprising how little is known about its basic properties in the multivariate case. The book therefore starts by exploring its fundamental properties and its limitations. The main part of the book is devoted to a complete and detailed elaboration of the new technique. A chapter with an extensive selection of finite elements follows as well as a chapter with formulas for Vandermonde determinants. Finally, the technique is applied to non-standard interpolations. The book is principally oriented to specialists in the field. However, since all the proofs are presented in full detail and since examples are profuse, a wider audience with a basic knowledge of analysis and linear algebra will draw profit from it. Indeed, the fundamental nature of multivariate nature of multivariate interpolation is reflected by the fact that readers coming from the disparate fields of algebraic geometry (singularities of surfaces), of finite elements and of CAGD will also all find useful information here.

### Keywords

Hermite Interpolation approximation theory finite element method multivariate polynomial

### Bibliographic information

• DOI https://doi.org/10.1007/BFb0088788
• Copyright Information Springer-Verlag Berlin Heidelberg 1992
• Publisher Name Springer, Berlin, Heidelberg
• eBook Packages
• Print ISBN 978-3-540-55870-5
• Online ISBN 978-3-540-47300-8
• Series Print ISSN 0075-8434
• Series Online ISSN 1617-9692
• Buy this book on publisher's site