Table Of ContentHausdorff Approximations
Mathematics and Its Applications (East European Series)
Managing Editor:
M. HAZEWINKEL
Centre for Mathematics and Computer Science, Amsterdam, The Netherlands
Editorial Board:
A. BIALY NICKI-BIRULA, Institute of Mathematics, Warsaw University, Poland
H. KURKE, Humboldt University, Berlin, G.D.R.
J. KURZWEIL, Mathematics Institute, Academy of Sciences, Prague, Czechoslovakia
L. LEINDLER, Bolyai Institute, Szeged, Hungary
L. LOVA sz, Bolyai Institute, Szeged, Hungary
D. S. MITRINOVIC, University of Belgrade, Yugoslavia
S. ROLEWICZ, Polish Academy of Sciences, Warsaw, Poland
BL. H. SENDOV, Bulgarian Academy of Sciences, Sofia, Bulgaria
I. T. TODOROV, Bulgarian Academy of Sciences, Sofia, Bulgaria
H. TRIEBEL, University ofl ena, G.D.R.
Volume 50
Hausdorff Approximations
by
BI. Sendov
Institute/or Mathematics,
Bulgarian Academy 0/ Sciences, Bulgaria
edited by
Gerald Beer
KLUWER ACADEMIC PUBLISHERS
DORDRECHT / BOSTON / LONDON
Library of Congress Cataloging-in-Publication Data
Sandov. B 1 agavest.
LKhausdoc-ovye priblizhenlla. Engllshl
Hausdorf; approxlrnatlons ! by BI. Sendov.
p. c •. -- (Mathematlcs and ltS appllcatlons. East European
serles ; v. 50)
Translatlon of: Khausdorfov}'e pribllzheni ia.
:incluDes blbllographlcal references and lndexes.
1. Topolog1cal spaces. 2. Aporoxlmatlon theory. I. Title.
I!. Serles Mathematlcs and ltS appl1catlons (Kluwer Academic
~ubllShers). Eas: Ewrcpean series; v. 50.
CA611.3.S4613 1990
514' .3--ac2C 90-43682
ISBN-13: 978-94-010-6787-4 e-ISBN-13: 978-94-009-0673-0
DOl: 10.1007/978-94-009-0673-0
Published by Kluwer Academic Publishers,
P.O. Box 17, 3300 AA Dordrecht, The Netherlands.
Kluwer Academic Publishers incorporates
the publishing programmes of
D. Reidel, Martinus Nijhoff, Dr W. Junk and MTP Press.
Sold and distributed in the U.S.A. and Canada
by Kluwer Academic Publishers,
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In all other countries, sold and distributed
by Kluwer Academic Publishers Group,
P.O. Box 322, 3300 AH Dordrecht, The Netherlands.
Printed on acid-free paper
© 1990 Bl. Sendov
Softcover reprint of the hardcover 1st edition 1990
All rights reserved, and available c/o JUSAUTOR, Sofia
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,
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SERIES EDITOR'S PREFACE
'Et moi, ... , si j'avait su comment en revenir, One service mathematics has rendered the
je n'y serais point a1Ie.' human race. It has put common sense back
Jules Verne where it belongs, on the topmost shelf next
to the dusty canister labelled 'discarded non
The series is divergent; therefore we may be sense'.
able to do something with it. Eric T. Bell
O. Heaviside
Mathematics is a tool for thought. A highly necessary tool in a world where both feedback and non
linearities abound. Similarly, all kinds of parts of mathematics serve as tools for other parts and for
other sciences.
Applying a simple rewriting rule to the quote on the right above one finds such statements as:
'One service topology has rendered mathematical physics .. .'; 'One service logic has rendered com
puter science .. .'; 'One service category theory has rendered mathematics .. .'. All arguably true. And
all statements obtainable this way form part of the raison d'etre of this series.
This series, Mathematics and Its Applications, started in 1977. Now that over one hundred
volumes have appeared it seems opportune to reexamine its scope. A t the time I wrote
"Growing specialization and diversification have brought a host of monographs and
textbooks on increasingly specialized topics. However, the 'tree' of knowledge of
mathematics and related fields does not grow only by putting forth new branches. It
also happens, quite often in fact, that branches which were thought to be completely
disparate are suddenly seen to be related. Further, the kind and level of sophistication
of mathematics applied in various sciences has changed drastically in recent years:
measure theory is used (non-trivially) in regional and theoretical economics; algebraic
geometry interacts with physics; the Minkowsky lemma, coding theory and the structure
of water meet one another in packing and covering theory; quantum fields, crystal
defects and mathematical programming profit from homotopy theory; Lie algebras are
relevant to filtering; and prediction and electrical engineering can use Stein spaces. And
in addition to this there are such new emerging subdisciplines as 'experimental
mathematics', 'CFD', 'completely integrable systems', 'chaos, synergetics and large-scale
order', which are almost impossible to fit into the existing classification schemes. They
draw upon widely different sections of mathematics."
By and large, all this still applies today. It is still true that at first sight mathematics seems rather
fragmented and that to find, see, and exploit the deeper underlying interrelations more effort is
needed and so are books that can help mathematicians and scientists do so. Accordingly MIA will
continue to try to make such books available.
If anything, the description I gave in 1977 is now an understatement. To the examples of
interaction areas one should add string theory where Riemann surfaces, algebraic geometry, modu
lar functions, knots, quantum field theory, Kac-Moodyalgebras, monstrous moonshine (and more)
all come together. And to the examples of things which can be usefully applied let me add the topic
'finite geometry'; a combination of words which sounds like it might not even exist, let alone be
applicable. And yet it is being applied: to statistics via designs, to radar/sonar detection arrays (via
finite projective planes), and to bus connections of VLSI chips (via difference sets). There seems to
be no part of (so-called pure) mathematics that is not in immediate danger of being applied. And,
accordingly, the applied mathematician needs to be aware of much more. Besides analysis and
numerics, the traditional workhorses, he may need all kinds of combinatorics, algebra, probability,
and so on.
In addition, the applied scientist needs to cope increasingly with the nonlinear world and the
vi SERIES EDITOR'S PREFACE
extra mathematical sophistication that this requires. For that is where the rewards are. Linear
models are honest and a bit sad and depressing: proportional efforts and results. It is in the non
linear world that infinitesimal inputs may result in macroscopic outputs (or vice versa). To appreci
ate what I am hinting at: if electronics were linear we would have no fun with transistors and com
puters; we would have no TV; in fact you would not be reading these lines.
There is also no safety in ignoring such outlandish things as nonstandard analysis, superspace
and anticommuting integration, p-adic and ultrametric space. All three have applications in both
electrical engineering and physics. Once, complex numbers were equally outlandish, but they fre
quently proved the shortest path between 'real' results. Similarly, the first two topics named have
already provided a number of 'wormhole' paths. There is no telling where all this is leading -
fortunately.
Thus the original scope of the series, which for various (sound) reasons now comprises five sub
series: white (Japan), yellow (China), red (USSR), blue (Eastern Europe), and green (everything
else), still applies. It has been enlarged a bit to include books treating of the tools from one subdis
cipline which are used in others. Thus the series still aims at books dealing with:
- a central concept which plays an important role in several different mathematical and/or
scientific specialization areas;
- new applications of the results and ideas from one area of scientific endeavour into another;
- infiuences which the results, problems and concepts of one field of enquiry have, and have had,
on the development of another.
Approximation of functions (by entities which can be coded in a finite way) is a topic of great
importance, both theoretically and computationally. For instance, uniform approximation by poly
nomials on intervals, or various approximation methods where the error is measured by some
integral (L2, V, ... ), all subjects with a considerable literature.
However, when the function to be approximated is discontinuous, and many natural phenomena
do involve modelling by discontinuous functions, the approximation schemes have disadvantages, as
noted by Kolmogorov who, in fact, formulated some natural desirable 'axioms' in that setting.
Hausdorff approximation, which is defined by means of the Hausdorff distance between (com
pleted) graphs of the functions involved, satisfies these axioms (and more).
In the seventies, a great deal of work was done on Hausdorff approximation, predominantly by
Bulgarian mathematicians including, in particular, the author of the present volume. The original
Russian version of this unique book appeared in 1978. Surprisingly, it was never translated. There
fore, it is a pleasure to now welcome a translation of this important book, updated to include refer
ences to the published literature up to and including 1988.
The shortest path between two truths in the Never lend books, for no one ever returns
real domain passes through the compleJl them; the only books I have in my library
domain. are books that other folk have lent me.
J. Hadamard Anatole France
La physique ne nous donne pas seu1ement The function of an eJlpert is not to be more
l'o ccasion de resoudre des probU:mes ... e1le right than other people, but to be wrong for
nous fait pressentir 1a solution. more sophisticated reasons.
H. Poincare David Butler
Amsterdam, July 1990 Michiel Hazewinkel
Table of Contents
Series editor's preface v
Preface xi
Preface to the Russian edition xiii
Introduction xv
Chapter 1 Elements of segment analysis 1
§ 1.1. Segment arithmetic 1
1.1.1. Partial orderings 2
1.1.2. Lattice operations 2
1.1.3. Arithmetic operations 4
1.1.3.1. Addition and subtraction 4
1.1.3.2. Multiplication and division 5
1.1.4. Distance and nonn 7
§ 1.2. Segment sequences 8
1.2.1. Segment limits 8
1.2.2. Theorems on segment limits 10
§ 1.3. Segment functions 12
1.3.1. The segment limit of a segment function 13
1.3.2. Segment derivatives 15
1.3.3. Segment continuity 17
1.3.4. H-continuity 18
Chapter 2 Hausdorff distance 23
§ 2.1. Hausdorff distance between subsets of a metric space 23
§ 2.2. The metric space FA 25
§ 2.3. H-distance in Ail and its properties 31
§ 2.4. Relationships between unifonn distance and the Hausdorff distance 35
viii TABLE OF CONTENTS
§ 2.5. The modulus of H-continuity 40
§ 2.6. The order of the modulus of H-continuity 42
§ 2.7. H-continuity on a subset 44
§ 2.8. H-distance with weight 46
Chapter 3 Linear methods of approximation 49
§ 3.1. Convergence of sequences of positive operators 49
§ 3.2. The order of approximation of functions by positive linear operators 54
§ 3.3. Approximation of periodic functions by positive integral operators 55
3.3.1. TheFejeroperator 57
3.3.2 The Jackson operator 61
3.3.3. The generalized Jackson operator 64
3.3.4 The Vall6e-Poussin operator 67
§ 3.4. Approximation of functions by positive integral operators on a finite closed
interval 70
3.4.1. The Landau operator 70
3.4.2. The generalized Landau operator 71
§ 3.5. Approximation of functions by summation formulas on a finite closed interval 71
3.5.1. Bernstein polynomials 72
3.5.2. Fejer interpolational polynomials 78
§ 3.6. Approximation of nonperiodic functions by integral operators on the entire
real axis 81
3.6.1 The Fejer operator in the nonp eriodic case 82
3.6.2. The generalized Jackson operator in the nonperiodic case 83
3.6.3. The Weierstrass operator 84
§ 3.7. Convergence of derivatives of linear operators 85
§ 3.8. A-distance 96
§ 3.9. Approximation by partial sums of Fourier series 99
Chapter 4 Best Hausdorff approximations 108
§ 4.1. Best approximation by algebraic and trigonometric polynomials 109
4.1.1. Uniqueness conditions for the polynomial of best approximation 112
4.1.2. Estimates for the best approximation 120
4.1.2.1. Bestapproximation of the delta-function 120
4.1.2.2. Universal estimates 125
4.1.2.3. Exact asymptotic behavior of the best approximation 128
TABLE OF CONTENTS ix
4.1.2.4. Generalizations of Jackson's theorem 140
4.1.2.5. Approximation of certain concrete functions 142
4.1.2.6. Approximation of convex functions 157
4.1.2.7. An analogue of Nikol'skii's theorem 165
4.1.2.8. Comonotone approximations 167
§ 4.2. Bestapproximation by mtional functions 174
4.2.1. Universal estimates for bounded functions 175
4.2.2. Unimprovability of the universal estimate 190
4.2.3. Approximation of analytic functions with singularities on the boundary
of a closed interval 198
§ 4.3. Best approximation by spline functions 203
4.3.1. Spline functions with equidistant knots 206
4.3.2. Spline functions with free knots 212
§ 4.4. Best approximation by piecewise monotone functions 213
Chapter 5 Converse theorems 227
§ 5.1. Existence of a function with preassigned best approximations 227
§ 5.2. Converse theorems for the approximation by algebmic and trigonometric
polynomials 233
5.2.1. The trigonometric case 234
5.2.2. The algebraic case 244
§ 5.3. Converse theorems for approximation by spline functions 247
§ 5.4. Converse theorems for approximation by mtional and partially monotone
functions 255
§ 5.5. Converse theorems for approximation by positive linear opemtors 257
Chapter 6 e-Entropy, e-capacity and widths 263
§ 6.1. e-entropy and e-capacity of the set F~ 264
§ 6.2. The number of (p,q)-corridors 270
§ 6.3. Labyrinths 280
6.3.1. Passages in labyrinths 289
§ 6.4. e-entropy and e-capacity of bounded sets of connected compact sets 291
§ 6.5. Widths 293
6.S.1. Widths of the set of bounded real functions 293
x TABLE OF CONTENTS
Chapter 7 Approximation of curves and compact sets in the plane 304
§ 7.1. Approximation by polynomial curves 308
§ 7.2. Characterization of best approximation in tenns of metric dimension 315
§ 7.3. Approximation by piecewise monotone curves 318
§ 7.4. Other methods for the approximation of curves in the plane 319
Chapter 8 Numerical methods of best Hausdorff approximation 322
§ 8.1. One-sided Hausdorff distance 322
8.1.1. Existence and uniqueness of the polynomial of best one
sided approximation 323
§ 8.2. Coincidence of polynomials of best approximation with respect to
one-and two-sided Hausdorff distance 326
§ 8.3. Numerical methods for calculating the polynomial of best one-sided
approximation 327
References 333
Author Index 355
Notation Index 361
Subject Index 363
