Multivariate approximation

551 Pages·2018·2.14 MB·English

by  Temlyakov

CAMBRIDGE MONOGRAPHS ON
APPLIED AND COMPUTATIONAL
MATHEMATICS
SeriesEditors
M. ABLOWITZ, S. DAVIS, J. HINCH,
A. ISERLES, J. OCKENDON, P. OLVER
32 Multivariate Approximation
TheCambridgeMonographsonAppliedandComputationalMathematicsseriesreflectsthecrucial
role of mathematical and computational techniques in contemporary science. The series publishes
expositionsonallaspectsofapplicableandnumericalmathematics,withanemphasisonnewdevel-
opmentsinthisfast-movingareaofresearch.
State-of-the-artmethodsandalgorithmsaswellasmodernmathematicaldescriptionsofphysical
andmechanicalideasarepresentedinamannersuitedtograduateresearchstudentsandprofessionals
alike.Soundpedagogicalpresentationisaprerequisite.Itisintendedthatbooksintheserieswillserve
toinformanewgenerationofresearchers.
Acompletelistofbooksintheseriescanbefoundat
www.cambridge.org/mathematics.
Recenttitlesincludethefollowing:
17. Scattereddataapproximation,HolgerWendland
18. Moderncomputerarithmetic,RichardBrent&PaulZimmermann
19. Matrixpreconditioningtechniquesandapplications,KeChen
20. Greedyapproximation,VladimirTemlyakov
21. Spectralmethodsfortime-dependentproblems,JanHesthaven,SigalGottlieb&DavidGottlieb
22. Themathematicalfoundationsofmixing,RobSturman,JulioM.Ottino&StephenWiggins
23. Curveandsurfacereconstruction,TamalK.Dey
24. Learningtheory,FelipeCucker&DingXuanZhou
25. Algebraicgeometryandstatisticallearningtheory,SumioWatanabe
26. Apracticalguidetotheinvariantcalculus,ElizabethLouiseMansfield
27. Differenceequationsbydifferentialequationmethods,PeterE.Hydon
28. MultiscalemethodsforFredholmintegralequations,ZhongyingChen,CharlesA.Micchelli&
YueshengXu
29. Partialdifferentialequationmethodsforimageinpainting,Carola-BibianeScho¨nlieb
30. Volterraintegralequations,HermannBrunner
31. Symmetry,phasemodulationandnonlinearwaves,ThomasJ.Bridges
32. Multivariateapproximation,V.Temlyakov
Multivariate Approximation
V. TEMLYAKOV
UniversityofSouthCarolina,
SteklovInstituteofMathematics,Moscow
and
LomonosovMoscowStateUniversity
UniversityPrintingHouse,CambridgeCB28BS,UnitedKingdom
OneLibertyPlaza,20thFloor,NewYork,NY10006,USA
477WilliamstownRoad,PortMelbourne,VIC3207,Australia
314–321,3rdFloor,Plot3,SplendorForum,JasolaDistrictCentre,NewDelhi–110025,India
79AnsonRoad,#06–04/06,Singapore079906
CambridgeUniversityPressispartoftheUniversityofCambridge.
ItfurtherstheUniversity’smissionbydisseminatingknowledgeinthepursuitof
education,learning,andresearchatthehighestinternationallevelsofexcellence.
www.cambridge.org
Informationonthistitle:www.cambridge.org/9781108428750
DOI:10.1017/9781108689687
©V.Temlyakov2018
Thispublicationisincopyright.Subjecttostatutoryexception
andtotheprovisionsofrelevantcollectivelicensingagreements,
noreproductionofanypartmaytakeplacewithoutthewritten
permissionofCambridgeUniversityPress.
Firstpublished2018
PrintedandboundinGreatBritainbyClaysLtd,ElcografS.p.A.
AcataloguerecordforthispublicationisavailablefromtheBritishLibrary.
ISBN978-1-108-42875-0Hardback
CambridgeUniversityPresshasnoresponsibilityforthepersistenceoraccuracy
ofURLsforexternalorthird-partyinternetwebsitesreferredtointhispublication
anddoesnotguaranteethatanycontentonsuchwebsitesis,orwillremain,
accurateorappropriate.
Contents
Preface pageix
1 ApproximationofUnivariateFunctions 1
1.1 Introduction 1
1.2 TrigonometricPolynomials 6
1.3 The Bernstein–Nikol’skii Inequalities. The Marcienkiewicz
Theorem 17
1.4 ApproximationofFunctionsintheClassesWr andHr 25
q,α q
1.5 HistoricalRemarks 34
2 OptimalityandOtherPropertiesoftheTrigonometricSystem 36
2.1 TheWidthsoftheClassesWr andHr 36
q,α q
2.2 FurtherPropertiesoftheTrigonometricSystem 54
2.3 ApproximationofFunctionswithInfiniteSmoothness 62
2.4 SamplingandNumericalIntegration 73
2.5 HistoricalRemarks 79
3 ApproximationofFunctionsfromAnisotropicSobolev
andNikol’skiiClasses 81
3.1 Introduction 81
3.2 TrigonometricPolynomials 82
3.3 TheBernstein–Nikol’skiiInequalitiesandTheir
Applications.AGeneralizationofthe
MarcinkiewiczTheorem 89
3.4 ApproximationofFunctionsintheClassesWr andHr 104
q,α q
3.5 EstimatesoftheWidthsoftheSobolevandNikol’skiiClasses 113
3.6 SamplingandNumericalIntegration 121
3.7 HistoricalRemarks 126
vi Contents
4 HyperbolicCrossApproximation 129
4.1 Introduction 129
4.2 SomeSpecialPolynomialswithHarmonicsin
HyperbolicCrosses 138
4.3 TheBernstein–Nikol’skiiInequalities 151
4.4 ApproximationofFunctionsintheClassesWr andHr 162
q,α q
4.5 SomeFurtherRemarks 186
4.6 HistoricalComments 189
4.7 OpenProblems 190
5 TheWidthsofClassesofFunctionswithMixedSmoothness 191
5.1 Introduction 191
5.2 TheOrthowidthsoftheClassesWr andHr 193
q,α q
5.3 TheKolmogorovWidthsoftheClassesWr andHr 216
q,α q
5.4 UniversalityofApproximationbyTrigonometric
PolynomialsfromtheHyperbolicCrosses 231
5.5 HistoricalRemarks 241
5.6 OpenProblems 242
6 NumericalIntegrationandApproximateRecovery 244
6.1 Introduction 244
6.2 CubatureFormulasandDiscrepancy 246
6.3 OptimalCubatureFormulasandNonlinearApproximation 253
6.4 LowerEstimates 262
6.5 TheFibonacciCubatureFormulas 272
6.6 TheKorobovCubatureFormulas 284
6.7 TheFrolovCubatureFormulas 289
6.8 UniversalCubatureFormulas 302
6.9 RecoveryofFunctions 305
6.10 HistoricalNotes,Comments,andSomeOpenProblems 315
6.11 OpenProblems 320
7 Entropy 321
7.1 Introduction.DefinitionsandSomeSimpleProperties 321
7.2 Finite-DimensionalSpaces.VolumeEstimates 323
7.3 SomeSimpleGeneralInequalities 325
7.4 AnInequalityBetweenEntropyNumbersandBest m-Term
Approximations 328
7.5 VolumeEstimatesforBallsofTrigonometricPolynomials 333
7.6 EntropyNumbersoftheBallsofTrigonometricPolynomials 345
7.7 EntropyNumbersfortheW-TypeFunctionClasses 363
Contents vii
7.8 EntropyNumbersfortheH-TypeFunctionClasses 373
7.9 DiscussionandOpenProblems 380
7.10 SomeHistoricalComments 383
8 GreedyApproximation 387
8.1 Introduction 387
8.2 TheTrigonometricSystem 394
8.3 WaveletBases 398
8.4 SomeInequalitiesfortheTensorProductofGreedyBases 404
8.5 Weight-GreedyBases 412
8.6 TheWeakChebyshevGreedyAlgorithm 415
8.7 SparseApproximationWithRespecttoGeneralDictionaries 422
8.8 OpenProblems 447
9 SparseApproximation 449
9.1 Introduction 449
9.2 ConstructiveSparseTrigonometricApproximation 454
9.3 ConstructiveSparseTrigonometricApproximationforSmall
Smoothness 472
9.4 OpenProblems 495
9.5 ConcludingRemarks 496
Appendix ClassicalInequalities 500
References 520
Index 532

Preface
The twentieth century was a period of transition from univariate problems
(i.e.,single-variableproblems) to multivariate problems in a number of areas
of mathematics. In many cases this step brought not only new phenomena
but also required new techniques. In some cases even the formulation of a
multivariateproblemrequiresanontrivialmodificationofaunivariateproblem.For
instance, the problem of the convergence of the multivariate trigonometric series
immediately encounters the question of which partial sums we should consider;
thereisnonaturalorderinginthemultivariatecase.Inotherwords:whatisanatural
multivariate analog of the trigonometric polynomials? In answering this question
mathematicians have studied different generalizations of the univariate trigono-
metricpolynomials:thosewithfrequenciesfromaball,acube,ahyperboliccross.
Multivariate problems turn out to be much more difficult than their univariate
counterparts.Themaingoalofthisbookistodemonstratetheevolutionoftheoret-
ical techniques from the univariate case to the multivariate case. We do this using
the example of the approximation of periodic functions. It is justified historically
andalsoitallowsustopresenttheideasinaconciseandclearway.Inmanycases
theseideascanbesuccessfullyusedinthenonperiodiccaseaswell.
We concentrate on a discussion of some theoretical problems which are impor-
tantinnumericalcomputation.Thefundamentalproblemofapproximationtheory
is to resolve a possibly complicated function, called the target function, into sim-
pler, easier to compute, functions called approximants. Generally, increasing the
resolutionofthetargetfunctioncanbeachievedonlybyincreasingthecomplexity
of the approximants. The understanding of this tradeoff between resolution and
complexityisthemaingoalofapproximationtheory.Thusthegoalsofapproxima-
tiontheoryandnumericalcomputationaresimilar,eventhoughapproximationthe-
ory is less concerned with computational issues. Approximation and computation
are intertwined and it is impossible to understand fully the possibilities in numer-
ical computation without a good understanding of the elements of approximation