Approximation by Multivariate Singular Integrals PDF
Preview Approximation by Multivariate Singular Integrals
SpringerBriefs in Mathematics Forfurthervolumes: http://www.springer.com/series/10030 George A. Anastassiou Approximation by Multivariate Singular Integrals 123 GeorgeA.Anastassiou DepartmentofMathematicalSciences UniversityofMemphis Memphis,TN38152 USA [email protected] ISSN2191-8198 e-ISSN2191-8201 ISBN978-1-4614-0588-7 e-ISBN978-1-4614-0589-4 DOI10.1007/978-1-4614-0589-4 SpringerNewYorkDordrechtHeidelbergLondon LibraryofCongressControlNumber:2011933230 MathematicsSubjectClassification(2010):41A17,41A25,41A28,41A35,41A36,41A60,41A80 ©GeorgeA.Anastassiou2011 Allrightsreserved.Thisworkmaynotbetranslatedorcopiedinwholeorinpartwithoutthewritten permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY10013, USA),except forbrief excerpts inconnection with reviews orscholarly analysis. Usein connectionwithanyformofinformationstorageandretrieval,electronicadaptation,computersoftware, orbysimilarordissimilarmethodologynowknownorhereafterdevelopedisforbidden. Theuseinthispublicationoftradenames,trademarks,servicemarks,andsimilarterms,eveniftheyare notidentifiedassuch,isnottobetakenasanexpressionofopinionastowhetherornottheyaresubject toproprietaryrights. Printedonacid-freepaper SpringerispartofSpringerScience+BusinessMedia(www.springer.com) To mywifeKoulaand mydaughters AngelaandPeggy Preface This short monograph is the first to deal exclusively with the study of the approximationofmultivariatesingularintegralsto the identity-unitoperator.Here westudyquantitativelythebasicapproximationpropertiesofthegeneralmultivari- ate singular integral operators, special cases of which are the multivariate Picard, Gauss-Weierstrass, Poisson-Cauchy and trigonometric singular integral operators, etc. These operators are not general positive linear operators. In particular we study the rate of convergence of these operators to the unit operator, as well as the related simultaneous approximation. These are given via inequalities and by the use of multivariate higher order modulus of smoothness of the high order partial derivatives of the involved function. Also we study the global smoothness preservationproperties of these operators. Some of these multivariate inequalities areprovedtobeattained,thatissharp.Furthermorewegiveasymptoticexpansions ofVoronovskayatypefortheerrorofapproximation.These propertiesare studied withrespecttoLpnorm,1 (cid:2)p (cid:2)1:ThelastchapterpresentsarelatedKorovkin type approximation theorem for functions of two variables. Plenty of examples aregiven. Fortheconvenienceofthereader,thechaptersareself-contained. Thisbriefmonographreliesonauthor’slasttwoyearsofrelatedresearchwork, morepreciselyseeauthor’sarticlesinthelistofreferencesofeachchapter. Advancedcoursescanbetaughtoutofthisshortbook.Allnecessarybackground andmotivationsaregivenperchapter. The presented results are expected to find applications in many areas of pure andappliedmathematics,suchasmathematicalanalysis,probability,statisticsand partial differential equations, etc. As such this brief monograph is suitable for researchers, graduate students, and seminars of the above subjects, also to be in allsciencelibraries. The preparation of this book took place during 2010–2011 in Memphis, Tennessee,USA. vii viii Preface Iwouldliketothankmyfamilyfortheirdedicationandlovetome,whichwas thestrongestsupportduringthewritingofthismonograph. DepartmentofMathematicalSciences GeorgeA.Anastassiou TheUniversityofMemphis,TN38152,USA Contents 1 UniformApproximationbyGeneralMultivariateSingular IntegralOperators............................................................ 1 1.1 Introduction .............................................................. 1 1.2 MainResults ............................................................. 1 1.3 Applications.............................................................. 14 References...................................................................... 16 2 Lp-Approximationby GeneralMultivariateSingular IntegralOperators............................................................ 19 2.1 Introduction .............................................................. 19 2.2 MainResults ............................................................. 19 2.3 Applications.............................................................. 30 References...................................................................... 32 3 Global Smoothness Preservation and Simultaneous ApproximationbyMultivariateGeneralSingularIntegrals............. 33 3.1 Introduction .............................................................. 33 3.2 MainResults ............................................................. 34 3.3 Applications.............................................................. 44 3.4 Conclusion ............................................................... 45 References...................................................................... 45 4 Multivariate VoronovskayaAsymptotic Expansions forGeneralSingularIntegrals .............................................. 47 4.1 Introduction .............................................................. 47 4.2 MainResults ............................................................. 47 4.3 Applications.............................................................. 55 References...................................................................... 56 ix
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