Methods of Fourier Analysis and Approximation Theory

255 Pages·2016·3.18 MB·English

by  Michael Ruzhansky

Applied and Numerical Harmonic Analysis
Michael Ruzhansky
Sergey Tikhonov
Editors
Methods of Fourier
Analysis and
Approximation
Theory

Applied and Numerical Harmonic Analysis
SeriesEditor
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UniversityofMaryland
CollegePark,MD,USA
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Moreinformationaboutthisseriesathttp://www.springer.com/series/4968
Michael Ruzhansky (cid:129) Sergey Tikhonov
Editors
Methods of Fourier Analysis
and Approximation Theory
Editors
MichaelRuzhansky SergeyTikhonov
DepartmentofMathematics ICREAResearchProfessor
ImperialCollegeLondon CentredeRecercaMatemaJtica
London,UnitedKingdom Barcelona,Spain
ISSN2296-5009 ISSN2296-5017 (electronic)
AppliedandNumericalHarmonicAnalysis
ISBN978-3-319-27465-2 ISBN978-3-319-27466-9 (eBook)
DOI10.1007/978-3-319-27466-9
LibraryofCongressControlNumber:2016932897
MathematicsSubjectClassification(2010):41-XX,42-XX32A-XX,65C-XX,49K-XX
©SpringerInternationalPublishingSwitzerland2016
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Preface
Thisvolumeconsistsofacollectionofpapersoriginatingfromtwoeventsdevoted
toareasofharmonicanalysisandtheirinterplaywiththeapproximationtheory.
The first event took place during the 9th ISAAC Congress in Krakow, Poland,
5–9August2013,atthesection“ApproximationTheoryandFourierAnalysis”.
The second event was the conference on Fourier Analysis and Approximation
Theory in the Centre de Recerca Matemàtica (CRM), Barcelona, during 4–8
November2013,organizedby the authors.Itcontinuedthe successfultraditionof
workshopseries: Osaka University 2008, ImperialCollege London2008, Nagoya
University2009,UniversityofGöttingen2010,ICREAConference2011inCRM,
andAaltoUniversity2012.
Thetopicsoftheconferenceinclude:Fourieranalysis,functionspaces,pseudo-
differential operators, microlocal and time-frequency analysis, partial differential
equations,andtheirlinkstomoderndevelopmentsintheapproximationtheory.
London,UK MichaelRuzhansky
Barcelona,Spain SergeyTikhonov
November2015
v

Contents
SomeProblemsinFourierAnalysisandApproximationTheory........... 1
MichaelRuzhanskyandSergeyTikhonov
PartI FourierAnalysis
ParsevalFrameswithnC1VectorsinRn .................................... 23
LauraDeCarliandZhongyuanHu
HyperbolicHardyClassesandLogarithmicBlochSpaces.................. 33
EvgueniDoubtsov
MultidimensionalExtremalLogan’sandBohman’sProblems............. 43
D.V.Gorbachev
WeightedEstimatesfortheDiscreteHilbertTransform..................... 59
E.Liflyand
Q-Measures on the Dyadic Group and Uniqueness Sets
forHaarSeries ................................................................... 71
MikhailG.Plotnikov
Off-Diagonal and Pointwise Estimates for Compact
Calderón-ZygmundOperators................................................. 85
PacoVillarroya
PartII FunctionSpacesofRadialFunctions
ElementaryProofsof Embedding TheoremsforPotential
SpacesofRadialFunctions...................................................... 115
PabloL.DeNápoliandIreneDrelichman
OnLeray’sFormula ............................................................. 139
E.LiflyandandS.Samko
vii
viii Contents
PartIII ApproximationTheory
Order of Approximationof Besov Classes in the Metric
ofAnisotropicLorentzSpaces.................................................. 149
K.A.Bekmaganbetov
AnaloguesofUlyanovInequalitiesforMixedModuliofSmoothness...... 161
M.K.PotapovandB.V.Simonov
ReconstructionOperatorofFunctionsfromtheSobolevSpace............ 181
N.T.Tleukhanova
PartIV OptimizationTheoryandRelatedTopics
Laplace–BorelTransformationofFunctionsHolomorphic
intheTorusandEquivalenttoEntireFunctions............................. 195
L.S.Maergoiz
Optimization Control Problems for Systems Described
byEllipticVariationalInequalitieswithStateConstraints.................. 211
SimonSerovajsky
Two ApproximationMethods of the FunctionalGradient
foraDistributedOptimizationControlProblem............................. 225
IlyasShakenov
NumericalModeling oftheLinear RelaxationalFiltration
byMonteCarloMethods........................................................ 237
KanatShakenov
Some Problems in Fourier Analysis
and Approximation Theory
MichaelRuzhanskyandSergeyTikhonov
Abstract We give a short overview of some questions and methods of Fourier
analysis, approximationtheory, and optimization theory that constitute an area of
currentresearch.
Keywords Approximation theory (cid:129) Fourier analysis (cid:129) Harmonic analysis (cid:129)
Optimizationtheory
MathematicsSubjectClassification 42-06,42-02,41-02,41-06
1 Introduction
Inthisreviewwepresentashortsurveyoftopicsrelatedtothetwoconferences(see
theprefacetothevolume),withbriefintroductiontomoreextensivepresentations
containedinthepapersofthiscollection.
2 Fourier Analysis
2.1 Parseval Frames
Chapter 1 by L. De Carli and Z. Hu is “Parseval Frames with n C 1 Vectors
in Rn”. A frame in a finite-dimensional vector space is a set of vectors that
M.Ruzhansky((cid:2))
DepartmentofMathematics,ImperialCollegeLondon,180Queen’sGate,LondonSW72AZ,
UK
e-mail:[e-mail]
S.Tikhonov
ICREA,CentredeRecercaMatemàtica,andUAB,CampusdeBellaterra,EdificiC,08193
Bellaterra(Barcelona),Spain
e-mail:[e-mail]
©SpringerInternationalPublishingSwitzerland2016 1
M.Ruzhansky,S.Tikhonov(eds.),MethodsofFourierAnalysisandApproximation
Theory,AppliedandNumericalHarmonicAnalysis,
DOI10.1007/978-3-319-27466-9_1