Multivariate Birkhoff Interpolation
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
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Lecture Notes in Mathematics Editors: A. Dold, Heidelberg B. Eckmann, ZUrich F. Takens, Groningen
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Rudolph A. Lorentz
Multivariate Birkhoff Interpolation
Springer-Verlag Berlin Heidelberg New York London Paris Tokyo Hong Kong Barcelona Budapest
Author Rudolph A. Lorentz Gesellschaft fur Mathematik und Datenverarbeitung Schlols Birlinghoven W-5205 St. Augustin 1, Germany and Universitat-Gesamthochschule-Duisburg Fachbereich 11, Mathematik Lotharstralse 63, W-4100 Duisburg, Germany
Mathematics Subject Classification (1991): 41A05, 41A63, 65D05, 65N30, 14517, 14117 ISBN 3-540-55870-5 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-55870-5 Springer-Verlag New York Berlin Heidelberg This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1992 Printed in Germany Typesetting: Camera ready by author/editor 46/3140-543210 - Printed on acid-free paper
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Contents 1 Introduction 2
Univariate Interpolation 2.1 Introduction and definitions 2.2 Main theorems .
3 Basic Properties of Birkhoff interpolation 3.1 Introduction and definitions 3.2 Properties of the spaces Ps 3.3 The Polya condition . . . . 3.4 Regular incidence matrices . 3.5 Properties of the determinant
1
4 4 6 9 9
13 16 17
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4 Singular Interpolation Schemes 4.1 Introduction and definitions . . . . . . . . . . . . . . . . . . . . . .. 4.2 Hermite interpolation of type total degree in lRd • • • . • • • . • •• , 4.3 Uniform Hermite interpolation of type total degree in lR2 , lR3 and lR4 4.4 Hermite interpolation of tensor-product type 4.5 Number-theoretic considerations. 4.6 Numerical results . . . . . . . 4.7 Slicing the pie the other way .
23 23 26 32 34 37 46 49
5 Shifts and Coalescences 5.1 Taylor expansion of the Vandermonde determinant 5.2 Definition of shifts 5.3 Existence of shifts . 5.4 Numbers of shifts . 5.5 Coefficients of the Taylor expansion 5.6 Coalescences .
50 50 50 52 55
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Decomposition Theorems 6.1 Introduction . 6.2 Decomposition theorems without knots 6.3 Decomposition theorems with nodes. 6.4 Comparison with other approaches
7 Reduction 7.1 Introduction . 7.2 The reduction theorem
57 60 62
62 62 64
68 72 72 72
CONTENTS
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8
Examples 8.1 Introduction . 8.2 Interpolation on rectangles 8.3 Triangular elements . . . .
75 75 78 86
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Uniform Hermite Interpolation of Tensor-product Type 9.1 Introduction . . . . 9.2 The Polya condition . . . . . . .. 9.3 Basic theorems . . . . . . . . . .. 9.4 Application of the basic theorems. 9.
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