Approximation Theory and Harmonic Analysis on Spheres and Balls

446 Pages·2013·2.65 MB·English

by  Dai

Springer Monographs in Mathematics
Forfurthervolumes:
http://www.springer.com/series/3733

Feng Dai • Yuan Xu
Approximation Theory
and Harmonic Analysis
on Spheres and Balls
123
FengDai YuanXu
DepartmentofMathematical DepartmentofMathematics
andStatisticalSciences UniversityofOregon
UniversityofAlberta Eugene,OR,USA
Edmonton,AB,Canada
ISSN1439-7382
ISBN978-1-4614-6659-8 ISBN978-1-4614-6660-4(eBook)
DOI10.1007/978-1-4614-6660-4
SpringerNewYorkHeidelbergDordrechtLondon
LibraryofCongressControlNumber:2013934217
MathematicsSubjectClassification:41Axx,42Bxx,42Cxx,65Dxx
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Preface
This book is written as an introduction to analysis on the sphere and on the ball,
anditprovidesacohesiveaccountofrecentdevelopmentsinapproximationtheory
and harmonic analysis on these domains. Analysis on the unit sphere appears as
partofFourieranalysis,in thestudyofhomogeneousspaces, andin severalfields
in applied mathematics, from numerical analysis to geoscience, and it has seen
increased activity in recent years. Its materials, however, are mostly scattered in
papersandsectionsofbooksthatcovermoregeneraltopics.Ourgoalsaretwofold.
Thefirstistoprovideaself-containedbackgroundforreaderswhoareinterestedin
analysisonthesphere.Thesecondis to givea completetreatmentofsome recent
advancesinapproximationtheoryandharmonicanalysisonthespheredevelopedin
thelastfifteenyearsorso,ofwhichbothauthorsareamongtheearnestparticipants,
andseveralchaptersofthebookarebasedonmaterialsfromtheirownresearch.
The book is loosely divided into four parts. The first part deals with analysis
on the sphere with respect to the surface measure dσ, the only rotation-invariant
measureonthesphere.Wegiveaself-containedexpositiononsphericalharmonics,
written with analysis in mind, in the first chapter, and present classical results of
harmonic analysis on the sphere, including convolutionstructure, Cesa`ro summa-
bility of orthogonal expansions, the Littlewood–Paley theory, and the multiplier
theorem due to Bonami–Clerc in the next two chapters. Approximation on the
sphere is discussed in the fourth section, where a recent characterization of best
approximation by polynomials on the sphere is given in terms of a modulus of
smoothnessanditsequivalentK-functional.Anintroductiontocubatureformulas,
which are necessary for discretizing integrals to obtain discrete processes of
approximation, is given in the sixth chapter. A recent proof of a conjecture on
spherical design, synonym of equal-weight cubature formulas, by Bondarenko,
Radchenko,and Viazovska, is included,for which the necessary ingredientof the
Marcinkiewicz–Zygmund inequality is established in the fifth chapter, where the
inequalityandseveralothersareestablishedforthedoublingweightonthesphere.
The second part discusses analysis in weighted spaces on the sphere. The
backgroundof this part is a far-reaching extension of spherical harmonics due to
C.Dunkl,inwhichtheroleoftheorthogonalgroupisreplacedbyafinitereflection
v
vi Preface
group, the measure dσ is replaced by h2κ(x)dσ, where hκ is a weight function
invariantunderareflectiongroupwithκbeingaparameter,andsphericalharmonics
arereplacedbyh-sphericalharmonicsassociatedwiththeDunkloperators,afamily
ofcommutingdifferential–differenceoperatorsthatreplacepartialderivatives.The
study of h-spherical harmonic expansions started about fifteen years ago. Many
deeper results in analysis were established only in the case of the group Zd,
2
for which hκ is given by hκ(x) = ∏di=1|xi|κi. In order to avoid heavy algebraic
preparations,we give a self-containedexpositionof Dunkl’s theory in the case of
Zd,whichiscomposedtohighlightitsparalleltothetheoryofsphericalharmonics.
2
Most results on ordinary spherical harmonic expansions can be extended to h-
spherical harmonic expansions, including finer Lp results on projection operators
and the Cesa`ro means, maximal functions, and multiplier theorem, as well as a
characterization of best approximation that was developed by many authors. We
give complete proofsof these results, which are more challengingthan proofsfor
classicalresultsfordσ,andinfact,insomecases,simplifythoseproofsforclassical
resultswhentheparametersκaresettozero.
Thethirdpartdealswithanalysisontheunitballandonthesimplex.Thereare
closerelationsbetweenanalysisonspheresandthatonballsofdifferentdimensions,
which enables us to utilize the results in the part two to developa parallel theory
for approximation theory and harmonic analysis on the unit ball. There is also a
connectionbetweenanalysisontheballandthatonthesimplex,whichcarriesmuch,
butnotall,ofanalysisontheballovertothesimplex.Theseresultsarecomposed
inparalleltothedevelopmentonthesphere.
The fourth part consists of one chapter, the last chapter of the book, which
discusses five topics related to the main theme of the book: highly localized
polynomialframes,distributionofnodesof positivecubature,positiveandstrictly
positivedefinite functions,asymptoticsofminimaldiscreteenergy,andcomputer-
izedtomography.
Analysisonthespherehasseenincreasedactivityinthepasttwodecades.There
are other related topicsthat we decided notto include,for examplescattered data
interpolation,applicationsofsphericalradialbasisfunctions(zonalfunctions),and
numericalor computationalanalysis on the sphere. These topics are more closely
related to the applied and computational branches of approximation theory. Our
choices, dictated by our own strengths and limitations, are those topics that are
closelyrelatedtothemaintheme—approximationtheoryandharmonicanalysis—
ofthisbook.
We keep the references in the text to a mininum and leave references and
historical remarks to the last section of each chapter, entitled “Notes and Further
Results,” where we also point out further results related to the materials in the
chapter.Some commonnotationand terminologyare givenin the preambleatthe
frontof the book, and there are two fairly detailed indexes:a subject index and a
symbolindex.
During the preparation of this book, we were granted a “Research in Team”
for a week at the Banff International Research Station and two months at the
Preface vii
Centre de Recerca Matema`tica, Barcelona, where we participated in the program
ApproximationTheoryandFourierAnalysis.Wearegratefultobothinstitutions.We
thankespeciallytheorganizer,SergeyTikhonov,oftheCRMprogramforhishelp
in arranging our visit. We also thank Professor Heping Wang, of Capital Normal
University,China,forhisassistanceinourproofofthearea-regulardecomposition
of the sphere. The first author is greatly indebted to Professor Zeev Ditzian for
his generous help and constant encouragement. The second author used the draft
of the book in a seminar course at the University of Oregon, and he thanks his
colleaguesMarcin Bownik and Karol Dziedziul(TechnicalUniversity of Gdansk,
Poland)andgraduatestudentsThomasBell,NathanPerlmutter,ChristopherShum,
David Steinberg, and Li-An Wang for keeping the course going. We thank our
editor,Kaitlin Leach, of Springer,for her professionaladvice and patience during
the preparationof our manuscript,and we thank the copy editor David Kramer at
Springer for numerousgrammaticaland stylish corrections. Finally, we gratefully
acknowledgethegrantsupportfromNSERCCanadaundergrantRGPIN 311678-
2010(F.D.)andtheNationalScienceFoundationundergrantDMS-1106113(Y.X.)
andagrantfromtheSimonsFoundation(#209057toYuanXu).
Edmonton,Canada FengDai
Eugene,OR YuanXu

To MyTeacher
ProfessorKunyangWang
F.D.
To Litian
WithAppreciation
Y.X.