Methods of Fourier Analysis and Approximation Theory PDF
Preview Methods of Fourier Analysis and Approximation Theory
Applied and Numerical Harmonic Analysis Michael Ruzhansky Sergey Tikhonov Editors Methods of Fourier Analysis and Approximation Theory Applied and Numerical Harmonic Analysis SeriesEditor JohnJ.Benedetto UniversityofMaryland CollegePark,MD,USA EditorialAdvisoryBoard AkramAldroubi GittaKutyniok VanderbiltUniversity TechnischeUniversitätBerlin Nashville,TN,USA Berlin,Germany DouglasCochran MauroMaggioni ArizonaStateUniversity DukeUniversity Phoenix,AZ,USA Durham,NC,USA HansG.Feichtinger ZuoweiShen UniversityofVienna NationalUniversityofSingapore Vienna,Austria Singapore,Singapore ChristopherHeil ThomasStrohmer GeorgiaInstituteofTechnology UniversityofCalifornia Atlanta,GA,USA Davis,CA,USA StéphaneJaffard YangWang UniversityofParisXII MichiganStateUniversity Paris,France EastLansing,MI,USA JelenaKovacˇevic´ CarnegieMellonUniversity Pittsburgh,PA,USA 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 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped. Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant protectivelawsandregulationsandthereforefreeforgeneraluse. Thepublisher,theauthorsandtheeditorsaresafetoassumethattheadviceandinformationinthisbook arebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsor theeditorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinorforany errorsoromissionsthatmayhavebeenmade. Printedonacid-freepaper ThisbookispublishedunderthetradenameBirkhäuser. TheregisteredcompanyisSpringerInternationalPublishingAGSwitzerland (www.birkhauser-science.com) 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:m.ruzhansky@imperial.ac.uk S.Tikhonov ICREA,CentredeRecercaMatemàtica,andUAB,CampusdeBellaterra,EdificiC,08193 Bellaterra(Barcelona),Spain e-mail:stikhonov@crm.cat ©SpringerInternationalPublishingSwitzerland2016 1 M.Ruzhansky,S.Tikhonov(eds.),MethodsofFourierAnalysisandApproximation Theory,AppliedandNumericalHarmonicAnalysis, DOI10.1007/978-3-319-27466-9_1
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