Rational Approximation of Real Functions

386 Pages·1988·11.97 MB·English

by  P. P. Petrushev

ENCYCLOPEDIA O F MATHEMATICS AN D ITS APPLICATION S
EDITED B Y G.-C.ROT A
Volume 28
Rational approximation of real function s

ENCYCLOPEDIA O F MATHEMATIC S AN D IT S APPLICATION S
Rational approximation
of real functions
P.P. PETRUSHEV, V.A . POPO V
Bulgarian Academy of Sciences
The right of the
University of Cambridge
to print and sell
all manner of books
was granted by
Henry VIII in 1534.
The University has printed
and published continuously
since 1584.
CAMBRIDGE UNIVERSIT Y PRES S
Cambridge
New York New Rochelle Melbourne Sydney
CCAAMMBBRRIDIDGGEE U UNNIVIVEERRSSITITYY P PRREESSSS
CCaammbbrrididggee, ,N Neeww Y Yoorkrk, ,M Meelblboouurrnnee, ,M Maaddrridid, ,C Caappee T Toowwnn, ,S Sininggaappoorere, ,
SSaaoo P Paauulolo, ,D Deelhlhii, ,D Duubbaaii, ,T Tookkyyoo, ,M Meexxicicoo C Citityy
CCaammbbrrididggee U Unniviveersrsitityy P Preresss s
TThhee E Eddininbbuurrgghh B Buuilidldiningg, ,C Caammbbrrididggee C CBB22 8 8RRUU, ,U UKK
PPuubblilsishheedd i inn t hthee U Unnitietedd S Statatetes so of fA Ammeerircicaa b byy C Caammbbrrididggee U Unniviveersrsitityy P Preressss, ,N Neeww Y Yoorrkk
www.cambridge.org
InInfoformrmaatitoionn o onn t hthiis st ittiltele:: www.cambridge.org/9780521331074
©© C Caammbbrrididggee U Unniviveersrsitityy P Preresss s1 1998877
TThhisi sp puubblilcicaatitoionn i is si inn c cooppyyrirgighhtt. .S Suubbjejecct tt too s statatututotorryy e exxcceepptitoionn
aanndd t too t hthee p prorovvisisioionns so of fr erelelevvaannt tc coolllelecctitvivee l ilciceennssiningg a aggrereeemmeenntsts, ,
nnoo r reepprroodduucctitoionn o of fa annyy p paarrt tm maayy t atakkee p plalaccee w witihthoouut tt hthee w wrirtittetenn
ppeermrmisisssioionn o of fC Caammbbrrididggee U Unniviveersrsitityy P Preressss. .
FFirisrst tp puubblilsishheedd 1 1998877
AA c caatatalologguuee r ereccoordrd f ofor rt hthisis p puubblilcicaatitoionn i sis a avvaailialabblele f rforomm t hthee B Brirtiitsihsh L Libibrararyry
LLibibrararyry o off C Coonnggreressss C Caatatalologguuiningg i nin P Puubblilcicaatitoionn d daatata
PPeetrturusshheevv, ,P P.P.P. .( P(Peennccoo P Peetrtorovv)), ,1 1994499--
RRaatitoionnaal la apppproroxximimaatitoionn o of fr ereaal lf ufunncctitoionnss. .
(E(Ennccyyccloloppeeddiaia o of fm maaththeemmaatitcics sa anndd i tits sa apppplilcicaatitoionnss; ;v v. .2 288) )
BBibibliloioggrarapphhyy
InInccluluddees si ninddeexx. .
11. .F Fuunncctitoionns so of fr ereaal lv vaariraiabbleless. .2 2 . .A Apppproroxximimaatitoionn t htheeooryry. .
I.I .P Pooppoovv, ,V Vaasisil lA A. .( V(Vaasisil lA Atatannaassoovv) )I II.I. T Titiltele. .
IIIII.I .S Seerireies s
QQAA333311.5.5.P.P4488 1 19988 77 5 51155.. 88 8 866-3-3443333 11
ISISBBNN 9 97788-0-0-5-52211-3-333110077--44 H Haarrddbbaacckk
CCaammbbrrididggee U Unniviveersrsitityy P Preresss sh haas sn noo r eressppoonnssibibiliiltityy f ofor rt hthee p peersrsisistetennccee o or r
aaccccuuraraccyy o of fU URRLLs sf ofor re exxteternrnaal lo or rt hthiridrd-p-paartrtyy i ninteternrneet tw weebbssitietes sr erefeferrreredd t too i inn
ththisi sp puubblilcicaatitoionn, ,a anndd d dooees sn noot tg guuaararannteteee t hthaat ta annyy c coonntetennt to onn s suucchh w weebbssitietes si sis, ,
oor rw wilill lr eremmaainin, ,a accccuurarattee o or ra apppproroppriraiatete. .I nInfoformrmaatitoionn r ereggaardrdiningg p prirciceess, ,t rtaravveel l
titmimeetatabbleless, ,a anndd o oththeer rf afacctutuaal li ninfoformrmaatitoionn g giviveenn i inn t hthiis sw woorrkk i is sc coorrrerecct ta at t
ththee t itmimee o of ff ifrisrtst p prrinintitningg b buut tC Caammbbrrididggee U Unniviveersrsitityy P Preresss sd dooees sn noot tg guuaararannteteee
ththee a accccuuraraccyy o of fs suucchh i ninfoformrmaatitoionn t htheerereaaftfeterr. .
∞
∞
CONTENTS

————————————————————————————
Preface ix
Acknowledgements xi
1 Qualitative theory of linear approximation 1
1.1 Approximation in normed linear spaces 1
1.2 Characterization of the algebraic polynomial of best uniform
approximation 6
1.3 Numerical methods 9
1.4 Notes 15
2 Qualitative theory of the best rational approximation 17
2.1 Existence 17
2.2 Uniqueness and characterization of the best uniform approximation 20
2.3 Nonuniqueness in L (a, b), 1 p < 26
p
2.4 Properties of the rational metric projection in 27
2.5 Numerical methods for best uniform approximation 33
2.6 Notes 37
3 Some classical results in the linear theory 40
3.1 Moduli of continuity and smoothness in C and L 40
p
3.2 Direct theorems: Jackson's theorem 46
3.3 Converse theorems 53
3.4 Direct and converse theorems for algebraic polynomial approximation
in a finite interval 57
3.5 Direct and converse theorems and the K-functional of J. Peetre 65
3.6 Notes 71
4 Approximation of some important functions 73
4.1 Newman's theorem 73
4.2 The exact order of the best uniform approximation to |x| 78
4.3 Zolotarjov's results 86
4.4 Uniform approximation of ex on [— 1,1]: Meinardus conjecture 94
4.5 Uniform approximation of e -x on [0, ) 100
4.6 Notes 105
vi Contents
5 Uniform approximation of some function classes 107
5.1 Preliminaries 107
5.2 Functions with rth derivative of bounded variation 114
5.3 Some classes of absolutely continuous functions and functions of bounded
variation 124
5.3.1. One technical result 124
5.3.2. Sobolev classes ,p> 1 128
5.3.3. Absolutely continuous functions with derivative in Orlicz space
Llog L 131
5.3.4. Functions with bounded variation and given modulus of continuity 134
5.4 DeVore's method 137
5.4.1. Hardy-Littlewood maximal function 137
5.4.2. More on the class W1,p> 1 141
p
5.4.3. Functions with derivative of bounded variation 144
5.5 Convex functions 146
5.6 Functions with singularities 153
5.7 Notes 158
6 Converse theorems for rational approximation 161
6.1 Gonchar's and Dol enko's results 161
6.2 Estimates for L -norms for the derivatives of rational functions and their
1
Hilbert transforms 164
6.3 Estimation for higher derivatives of rational functions and its applications 169
6.4 Notes 179
7 Spline approximation and Besov spaces 185
7.1 L (0<p< 1 ) spaces 185
p
7.2 Besov spaces 201
7.3 Spline approximation 203
7.3.1. Introduction 203
7.3.2. Direct and converse theorems in L (0 < p < ) 207
p
7.3.3. Direct and converse theorems in uniform metric and in BMO 219
7.4 Notes 223
8 Relations between rational and spline approximations 224
8.1 The rational functions are not worse than spline functions as a tool for
approximation in LL (( 11 pp << )) metric 224
pp
8.2 Estimate of spline approximation by means of rational in L , 11 << pp 240
p
8.3 Relations between rational and spline approximations of functions and
rational and spline approximations of their derivatives 243
8.4 Notes 260
9 Approximation with respect to Hausdorff distance 263
9.1 Hausdorff distance and its properties 264
9.2 Hausdorff approximation of the jump 266
9.3 Bounded functions 271
9.4 Notes 275
Contents vii
10 Theo-effect 278
10.1 Existence and characterization for uniform approximation of
individual functions of the class V and for L approximation of
r 1
functions of bounded variation 278
10.1.1. One new functional characteristic 279
10.1.2. The o-effect in some spline approximations 285
10.1.3. Uniform approximation of individual functions of the class V and
r
L approximation of functions of bounded variation 294
1
10.2 Uniform approximation of absolutely continuous functions 296
10.3 Uniform approximation of convex functions 300
10.4 L approximation of functions of bounded variation 302
p
10.5 Notes 304
11 Lower bounds 308
11.1 Some simple lower bounds 308
11.1.1. Negative results for uniform approximation of continuous functions
with given modulus of smoothness 308
11.1.2. One negative result for uniform approximation of absolutely con-
tinuous functions 311
11.1.3. Lower bound for uniform approximation of the functions of the
class V 312
r
11.1.4. Uniform approximation of convex functions 314
11.1.5. L approximation of functions of bounded variation 316
p
11.2 Uniform approximation of functions of bounded variation and given
modulus of continuity 317
11.3 Notes 328
12 Pade approximations 329
12.1 Definition and properties of the Pade approximants 329
12.2 Direct theorem for the rows of the Pade-table 332
12.3 Converse theorem for the rows of the Pade-table 336
12.4 The diagonal of the Pade-table 342
12.5 Notes 346
Appendix Some numerical results 348
References 357
Author index 368
Notation and subject index 370