Table Of ContentThe Theory of Cubature Fonnulas
Mathematics and Its Applications
Managing Editor:
M. HAZEWINKEL
Centrefor Mathematics anti Computer Science, Amsterdam, The Netherlands
Volume 415
The Theory of
Cubature Formulas
by
S. L. Sobolev t
and
V. L. Vaskevich
Sobolev Institute ofM athematics,
Siberian Division of the Russian Academy of Sciences,
Novosibirsk, Russia
Springer-Science+Business Media, B.Y.
A C.I.P. Catalogue record for this book is available from the Library of Congress.
ISBN 978-90-481-4875-2 ISBN 978-94-015-8913-0 (eBook)
DOI 10.1007/978-94-015-8913-0
This is a revised and updated translation
of the original Russian work of the same title.
©Sobolev Institute of Mathematics, Novosibirsk, 1996
Translated by S. S. Kutateladze.
Printed on acid-free paper
All Rights Reserved
© Springer Science+Business Media Dordrecht 1997
Originally published by Kluwer Academic Publishers in 1997.
Softcover reprint ofthe hardcover 1s t edition 1997
No part of the material protected by this copyright notice may be reproduced or
utilized in any form or by any means, electronic or mechanical,
including photocopying, recording or by any information storage and
retrieval system, without written permission from the copyright owner
Contents
Foreword to the English Translation ix
Preface xi
From the Preface to "Introduction to the Theory of Cubature
Formulas" xix
Chapter 1. Problems and Results of the Theory of Cubature
Formulas 1
§ 1. Exact Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
§ 2. Functional-Analytical Statement of the Problem . . . . . . . . . . . . . . . 7
§ 3. The Order of a Cubature Formula on Infinitely Differentiable
Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
wi
§ 4. Errors in m) ..............•..........................••.... 23
§ 5. Expansion of the L~m)* Norm of an Error with Arbitrary Nodes 33
§ 6. The Weights of Optimal Cubature Formulas on a Given Lattice 37
Chapter 2. Cubature Formulas of Finite Order 43
§ 1. Formulas of Interpolatory Type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
§ 2. Rotation Invariant Cubature Formulas 47
§ 3. Rotation Invariant Cubature Formulas on the Sphere in 1R3 64
VI Contents
Chapter 3. Formulas with Regular Boundary Layer for Rational
Polyhedra 74
§ 1. Rational Polyhedra 75
§ 2. Constructing Formulas for Rational Polyhedra 82
§ 3. A Formal Boundary Layer 86
Chapter 4. The Rate of Convergence of Cubature Formulas 93
§ 1. A Universal Lower Bound on the Rate of Convergence 93
§ 2. The Rate of Convergence of a Romogeneous Error ............. 103
§ 3. The Bakhvalov Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
§ 4. The Rate of Convergence of an Equidistributed Error 122
Chapter 5. Cubature Formulas with Regular Boundary Layer131
§ 1. The Properties of the Extremal Function of an L~m) -Optimal
Error ........................................................ 131
§ 2. Errors in the i~m)(n) Space of Compactly-Supported Functions 142
§ 3. Constructing a Formula with Regular Boundary Layer .. ...... 148
§ 4. Asymptotic Expansion of the Norm of an Error with Regular
Boundary Layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
§ 5. The Properties of the Extremal Function of an Error in L~m)(n) 164
Chapter 6. Universal Asymptotic Optimality 173
§ 1. Cubature Formulas with Bounded Boundary Layer
in Rilbert Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
§ 2. Cubature Formulas with Bounded Boundary Layer
in Rölder Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200
§ 3. Constructing Universal Asymptotically Optimal Formulas 211
Contents Vll
Chapter 7. Cubature Formulas of Infinite Order 221
§ 1. Weak Convergence of Cubature Formulas .................... . 221
§ 2. The Function Classes H(x,A,A) and C(X,A,A) .............. . 230
§ 3. The Properties of H(x,A) and C(x,A) for x ~ 1 235
§ 4. The Function Classes w(p, J.l) 239
(1,
H
§ 5. The Classes of Periodic Functions (x, A, A) 245
§ 6. Convergence of Cubature Formulas in H(x, A, A) 253
§ 7. Gevrey Classes of Functions in a Single Independent Variable 257
§ 8. Convergence of Euler-Maclaurin and Gregory Quadrature Formulas
on Gevrey Classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270
§ 9. The Sequence of the Fourier Coefficients of an Error . . . . . . . . . . . 275
§ 10. The Fourier Transform of a Local Error 280
§ 11. The Norm of the Error of a Gregory Quadrature Formula 288
Chapter 8. Functions of a Discrete Variable 292
§ 1. Operations over Discrete Functions 292
§ 2. Spaces of Discrete Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312
§ 3. The Fourier Transform of a Discrete Function . . . . . . . . . . . . . . . . . 324
Chapter 9. Optimal Formulas 331
§ 1. Statement of the Problem of Optimal Weights 331
§ 2. The Fundamental Solution to the Convolution Equation 337
§ 3. A Discrete Analog of the Polyharmonie Operator 344
§ 4. The Weights of Optimal Formulas and the Extension Problem 351
§ 5. A One-Dimensional Discrete Analog of a Derivative of Even
Order ....................................................... . 355
§ 6. The Roots of the Euler Polynomial ................... ......... 361
§ 7. The First Asymptotic Formula ............... ................. 368
§ 8. The Weights of Optimal Quadrature Formulas 375
References 389
Notation Index 411
Subject Index 413
Foreword to the English Translation
Academician Serge'f L' vovich Sobolev (1908-1989), a great Russian scholar and
the founder of the Institute of Mathematics of the Siberian Division of the Rus
sian Academy of Sciences at Novosibirsk which is now named after hirn, is world
renowned for his contribution to distribution theory, sharing the farne of its pro
pounding with L. Schwartz. S. L. Sobolev successfully applied his new functional
analytical technique not only to partial differential equations but also to computa
tional mathematics, changing the layout of the field of numerical integration.
The present edition is a translation of the posthumous monograph finished by
V. L. Vaskevich, the last pupil of S. L. Sobolev, and published in Russian by the
Sobolev Institute Press in 1996. The book contains all contributions of S. L. Sobolev
to numerical integration as well as results of his students and followers on cubature
formulas, thus granting an updated definitive source of this direction in modern
mathematics. For the first time, the book includes recent data about invariant
cubature formulas exact for spherical harmonics up to a given degree and Sobolev's
research on optimal cubature formulas and Euler polynomials.
This edition was typeset using AMS-TEX, the American Mathematical Soci
ety's TEX system.
S. L. Soholev was one of my inspired teachers in mathematics and life although
I have never plunged into numerical integration before. Translating the hook was
thus an imperative hut onerous duty since it covers the field of research deeply
rooted in classical mathematics as well as in the brand-new applied sections of
functional analysis. One technicality needs explanation, namely, absence of the
term distribution which is in common parlance in the West. The competing term
generalized function proliferates in Russia and is reverently retained as coined by
the inventor as far back as in the thirties of this century.
It gives me a blended feeling of great pleasure and deep sorrow to invite the
reader to get acquaintance with this masterpiece which is apart of the memory of
S. L. Sobolev to whom I am greatly indebted.
S. Kutateladze
Chapter 1
Problems and Results of the Theory of
Cubature Formulas
This chapter is introductory. Here we state the problems under study, outline
the principal ideas behind our theory and make an overview of the most impor
tant results. The intended conciseness of exposition results in making some proofs
schematic. The reader interested in more details may find them in the sequel.
§1 . Exact Formulas
The main problem of numerical integration consists in approximating the in
tegral J J
I('P) = 'P(x)dx = Xn(x)'P(x)dx. (1.1 )
n
Here x is an n-dimensional coordinate vector, and Xn (x) is the indicator of a con
n
nected domain with sufficiently smooth boundary. We seek for an approximant
by taking a linear combination of the values of 'P( x) at the N points
(1.2)
called nodes, namely,
J
LN LN
I*( 'P) = Ck'P(x(k») = Ckh(x - x(k»)'P(x) dx, (1.3)
k==l k==l
where h( x) is the conventional Dirae delta function. We call (1.3) a eubature formula
by analogy with a quadrature formula in the one-dimensional case.
