Hyperbolic Cross Approximation

222 Pages·2018·2.88 MB·English

by  Dinh Dũng

Advanced Courses in Mathematics CRM Barcelona
Dinh D ng
Vladimir Temlyakov
Tino Ullrich
Hyperbolic
Cross
Approximation

Advanced Courses in Mathematics
CRM Barcelona
Centre de Recerca Matemàtica
Managing Editor:
Enric Ventura

More information about this series at http://www.springer.com/series/5038
Dinh D(cid:458)ng • Vladimir Temlyakov • Tino Ullrich
Hyperbolic Cross Approximation
Editor for this volume:
Sergey Tikhonov, Centre de Recerca Matemàtica
Dinh D(cid:458)ng Vladimir Temlyakov
Information Technology Institute Department of Mathematics
Vietnam National University University of South Carolina
Hanoi, Vietnam Columbia, SC, USA

and
Tino Ullrich
Faculty of Mathematics Steklov Institute of Mathematics and
Technical University of Chemnitz Lomonosov Moscow State University
09126 Chemnitz, Germany Moscow, Russia

ISSN 2297-0304 ISSN 2297-0312 (electronic)
Advanced Courses in Mathematics - CRM Barcelona
ISBN 978-3-319-92239-3 ISBN 978-3-319-92240-9 (eBook)
https://doi.org/10.1007/978-3-319-92240-9

Library of Congress Control Number: 2018958719

Mathematics Subject Classification (2010): 41A15, 41A17, 41A46, 41A63, 42A05, 42A10, 42B05, 42B25,
42B99, 65D30, 65D32

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Contents
Foreword . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2 Trigonometric Polynomials. . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.1 Univariate polynomials . . . . . . . . . . . . . . . . . . . . . . . . 15
2.2 Multivariate polynomials . . . . . . . . . . . . . . . . . . . . . . . 18
2.3 Hyperbolic cross polynomials . . . . . . . . . . . . . . . . . . . . 21
2.4 The Bernstein–Nikol’skii inequalities . . . . . . . . . . . . . . . . 24
2.5 Volume estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.6 Riesz products and the small ball inequality . . . . . . . . . . . . 31
2.7 Comments and open problems . . . . . . . . . . . . . . . . . . . . 33
3 Function Spaces on Td . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.1 Spaces of functions with bounded mixed derivative . . . . . . . . 35
3.2 Spaces of functions with bounded mixed difference . . . . . . . . 37
3.3 Characterizationvia Fourier transform . . . . . . . . . . . . . . . 39
3.4 Embeddings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
4 Linear Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
4.2 Approximation by the hyperbolic cross polynomials . . . . . . . . 48
4.3 The Kolmogorovwidths . . . . . . . . . . . . . . . . . . . . . . . 54
4.4 Orthowidths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
4.5 The linear widths . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.6 Open problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
5 Sampling Recovery . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
5.1 The univariate de la Vall´ee Poussin interpolation . . . . . . . . . 75
5.2 Frequency-limited sampling representations –
discrete Littlewood–Paley . . . . . . . . . . . . . . . . . . . . . . 77
5.3 Sampling on the Smolyak grids . . . . . . . . . . . . . . . . . . . 79
5.4 Sampling widths . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
v
vi Contents
5.5 Time-limited sampling representations B-splines . . . . . . . . . . 87
5.6 Open problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
6 Entropy Numbers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
6.1 General notions and inequalities . . . . . . . . . . . . . . . . . . . 95
6.2 Entropy numbers for W classes in L . . . . . . . . . . . . . . . . 99
q
6.3 Entropy numbers for H and B classes in L . . . . . . . . . . . . 102
q
6.4 Entropy numbers and the small ball problem . . . . . . . . . . . 104
6.5 Open problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
7 Best m-Term Approximation . . . . . . . . . . . . . . . . . . . . . . . . 107
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
7.2 Orthogonal bases . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
7.3 Some related problems . . . . . . . . . . . . . . . . . . . . . . . . 113
7.4 A comment on Stechkin’s lemma . . . . . . . . . . . . . . . . . . 114
7.5 Sparse trigonometric approximation . . . . . . . . . . . . . . . . 116
7.6 Different types of greedy algorithms . . . . . . . . . . . . . . . . 124
7.7 Open problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
8 Numerical Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
8.1 The problem setting . . . . . . . . . . . . . . . . . . . . . . . . . 127
8.2 Lower bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
8.3 Cubature on Smolyak grids . . . . . . . . . . . . . . . . . . . . . 130
8.4 The Fibonacci cubature formulas . . . . . . . . . . . . . . . . . . 132
8.5 The Frolov cubature formulas . . . . . . . . . . . . . . . . . . . . 134
8.6 Modifications of Frolov’s method . . . . . . . . . . . . . . . . . . 138
8.7 Quasi Monte Carlo cubature . . . . . . . . . . . . . . . . . . . . . 140
8.8 Discrepancy and numerical integration . . . . . . . . . . . . . . . 143
8.9 Open problems and historical comments . . . . . . . . . . . . . . 150
9 Related Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
9.1 Why classes with mixed smoothness? . . . . . . . . . . . . . . . . 153
9.2 Universality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
9.3 Further generalizations . . . . . . . . . . . . . . . . . . . . . . . . 158
9.4 Direct and inverse theorems . . . . . . . . . . . . . . . . . . . . . 159
9.5 Kolmogorovwidths of the intersection of function classes . . . . . 159
9.6 Further s-numbers . . . . . . . . . . . . . . . . . . . . . . . . . . 161
9.7 The quasi Banach situation . . . . . . . . . . . . . . . . . . . . . 164
9.8 Sampling along lattices . . . . . . . . . . . . . . . . . . . . . . . . 167
9.9 Sampling recovery in energy norm . . . . . . . . . . . . . . . . . 168
9.10 Continuous algorithms in m-term approximation
and nonlinear widths . . . . . . . . . . . . . . . . . . . . . . . . . 169
Contents vii
10 High-Dimensional Approximation. . . . . . . . . . . . . . . . . . . . . . 171
10.1 Anisotropic mixed smoothness. . . . . . . . . . . . . . . . . . . . 171
10.2 Explicit constants and preasymptotics . . . . . . . . . . . . . . . 173
10.3 Approximation in the energy norm . . . . . . . . . . . . . . . . . 175
10.4 ε-dimension and approximation in infinite dimensions. . . . . . . 176
11 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
11.1 General notation . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
11.2 Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
11.3 Duality in L spaces . . . . . . . . . . . . . . . . . . . . . . . . . 183
p
11.4 Fourier series of functions in L . . . . . . . . . . . . . . . . . . . 184
p
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217
Foreword
This book contains expository lecture notes for the course given at the Advanced
CoursesonConstructiveApproximationandHarmonicAnalysis,whichtookplace
in the Centre de Recerca Matem`atica (CRM) in Bellaterra(Barcelona)May 30th
to June 3th, 2016.These courseswereone ofthe mainactivities ofthe five-month
Research Programme on Approximation Theory and Fourier Analysis.
Expanded versions of the lectures “Hyperbolic Cross Approximation” given
byVladimirTemlyakov(UniversityofSouthCarolina)andTinoUllrich(Technical
University of Chemnitz,Germany)are presented here.
We gather a survey on multivariate approximation. The 20th century was
a period of transition from univariate problems to multivariate problems in a
number of areas of mathematics. For instance, it is a step from Gaussian sums
to Weil’s sums in number theory, a step from ordinary differential equations to
PDEs, a step from univariate trigonometric series to multivariate trigonometric
seriesinharmonicanalysis,astepfromquadratureformulastocubatureformulas
in numerical integration, a step from univariate function classes to multivariate
functionclassesinapproximationtheory.Inmanycasesthisstepbroughtnotonly
new phenomena but also required new techniques to handle the corresponding
multivariateproblems.Insomecasesevenaformulationofamultivariateproblem
requiresa nontrivialmodification ofa univariate problem. Forinstance, the prob-
lemofconvergenceofthemultivariatetrigonometricseriesimmediatelyencounters
aquestionofwhichpartialsums weshouldconsider–there isno naturalordering
in the multivariate case. In other words: What is a natural multivariate analog
of the univariate trigonometric polynomials? Answering this questionmathemati-
ciansstudieddifferentgeneralizationsoftheunivariatetrigonometricpolynomials:
with frequencies from a ball, a cube or, most importantly, a hyperbolic cross.
Results discussedin this survey demonstratethat polynomials with frequen-
cies from hyperbolic crosses play the same role in the multivariate approximation
astheunivariatetrigonometricpolynomialsplayintheapproximationoffunctions
onasinglevariable.Resultsobtainedinthemultivariateapproximationtheoryfor
thelast50yearsestablishedadeepconnectionbetweentrigonometricpolynomials
with frequencies from the hyperbolic crossesand classes of functions defined with
the help ofeither mixed derivativesormixeddifferences. The importance ofthese
classes was understood in the beginning of the 1960s. The multivariate problems
ix
x Foreword
of hyperbolic cross approximation turn out to be much more involved than their
univariate counterparts.
Approximation of classes of functions with bounded mixed derivative and
functions with a restriction on mixed differences have been developed following
classical tradition. A systematic study of different asymptotic characteristics of
these classes dates back to the beginning of the 1960s. In the book the authors
considerboth linear approximationproblems – the problems of approximationby
elementsofagivenfinite-dimensionallinearsubspaceandnonlinearapproximation
problems. They discuss a number of the most important asymptotic characteris-
tics related to the linear approximation. In addition to the Kolmogorov n-width,
the authors also study the asymptotic behavior of the linear widths and the or-
thowidths.Inthe definitionoflinearwidthweallowalllinearoperatorsofrankm
to compete in the minimization problem.Clearly, we wouldlike to workwith nice
andsimple linear operators.This idea motivated researchersto impose additional
restrictions on the linear operators in the definitions of the corresponding modifi-
cationsofthe linearwidth. Oneverynaturalrestrictionisthatthe approximating
rank m operator is an orthogonal projection operator. This leads to the concept
ofthe orthowidth(Fourier width). Another natural restrictionis that the approx-
imating rank m operator is a recovering operator, which uses function values at
m points.
Recently, driven by applications in engineering, biology, medicine and other
areas of science, nonlinear approximation began to play an important role. Non-
linear approximation is important in applications because of its concise represen-
tations and increased computational efficiency. In the book the authors discuss a
typicalproblemofnonlinearapproximation–them-termapproximation.Another
name for m-term approximation is sparse approximation.
Discretizationisaveryimportantstepinmakingacontinuousproblemcom-
putationallyfeasible.Theproblemofconstructionofgoodsetsofpointsinamulti-
dimensionaldomainisafundamental problemofmathematicsandcomputational
mathematics.
Aprominentexampleofclassicaldiscretizationproblemisaproblemofmet-
ricentropy(coveringnumbers,entropynumbers).Theauthorsdiscussthisproblem
in detail. Bounds for the entropy of function classes are important by themselves
andalsohaveimportantconnectionstootherfundamentalproblems.Forinstance,
theproblemoftheentropyofsomeclassesoffunctionswithboundedmixedderiva-
tive is equivalent to the fundamental small ball problem from probability theory.
Another prominent example of a discretization problem, which is discussed
in the book, is the problem of numerical integration. It turns out that contrary
to the numerical integrationin the univariate case and in the multivariate case of
the anisotropic smoothness classes, where regular grids methods are optimal (in
the sense of order), in the case of numerical integration of functions with mixed
smoothnessthe regulargrids methods are veryfar frombeing optimal. Numerical
integration in the mixed smoothness classes requires deep number theoretical re-