Approximation by Multivariate Singular Integrals

90 Pages·2011·0.451 MB·English

by  George A. Anastassiou (auth.)

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,
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Printedonacid-freepaper
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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
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