Springer Proceedings in Mathematics Volume 13 Forfurthervolumes: http://www.springer.com/series/8806 Springer Proceedings in Mathematics This book series features volumes of selected contributions from workshops and conferencesinallareasofcurrentresearchactivityinmathematics.Afteranoverall evaluation,atthehandsofthepublisher,oftheinterest,scientificquality,andtime- linessofeachproposal,everyindividualcontributionhasbeenrefereedtostandards comparabletothoseofleadingmathematicsjournals.Thisseriesthuspresentstothe researchcommunitywell-editedandauthoritativereportsonnewestdevelopments inthemostinterestingandpromisingareasofmathematicalresearchtoday. Marian Neamtu • Larry Schumaker Editors Approximation Theory XIII: San Antonio 2010 123 Editors MarianNeamtu LarrySchumaker CenterforConstructiveApproximation CenterforConstructiveApproximation DepartmentofMathematics DepartmentofMathematics VanderbiltUniversity VanderbiltUniversity Nashville,TN37240 Nashville,TN37240 USA USA [email protected] [email protected] ISSN2190-5614 e-ISSN2190-5622 ISBN978-1-4614-0771-3 e-ISBN978-1-4614-0772-0 DOI10.1007/978-1-4614-0772-0 SpringerNewYorkDordrechtHeidelbergLondon MathematicsSubjectClassification(2010):41Axx,65Dxx (cid:2)c SpringerScience+BusinessMedia,LLC2012. 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) Preface These proceedings were prepared in connection with the international confer- ence Approximation Theory XIII, which was held during March 7–10, 2010 in San Antonio, Texas. The conference was the thirteenth in a series of meetings in ApproximationTheory held at variouslocations in the United States, and was at- tendedby144participants.PreviousconferencesintheserieswereheldinAustin, Texas(1973,1976,1980,1992);CollegeStation,Texas(1983,1986,1989,1995); Nashville, Tennessee (1998), St. Louis, Missouri (2001); Gatlinburg, Tennessee (2004);andSanAntonio,Texas(2007). Weareparticularlyindebtedtoourplenaryspeakers:AlbertCohen(Paris),Oleg Davydov(Strathclyde),GregoryFasshauer(IllinoisInstituteofTechnology),Anne Gibert (University of Michigan), Bin Han (University of Alberta), Kirill Kopotun (UniversityofManitoba),andVilmosTotik(UniversityofSouthFlorida),whopro- vided inspiring talks and set a high standard of expositionin their descriptionsof newdirectionsforresearch.Theconferencealsoprovidedaforumfortheawarding of the PopovPrize in ApproximationTheory.The sixth Vasil A. PopovPrize was awardedtoJoelA.Tropp(CalTech),whoalsopresentedaplenarylecture.Thanks are also due to the presenters of contributed papers, as well as everyone who at- tended,formakingtheconferenceasuccess. WeareespeciallygratefultotheNationalScienceFoundationforfinancialsup- port, and also to the Department of Mathematics at Vanderbilt University for its logisticalsupport. Wewouldalsoliketoexpressoursinceregratitudetothereviewerswhohelped selectarticlesforinclusioninthisproceedingsvolume,andalsofortheirsuggestions totheauthorsforimprovingtheirpapers. Nashville,TN MarianNeamtu LarryL.Schumaker v Contents An Asymptotic Equivalence Between Two Frame Perturbation Theorems ...................................................... 1 B.A.Bailey 1 ThePerturbationTheorems................................. 1 2 AnAsymptoticEquivalence ................................ 3 References..................................................... 7 GrowthBehaviorandZero Distribution ofMaximally Convergent RationalApproximants .......................................... 9 Hans-PeterBlatt,Rene´ Grothmann,andRalitzaK.Kovacheva References..................................................... 15 GeneralizationofPolynomialInterpolationatChebyshevNodes ........ 17 DebaoChen 1 Introduction.............................................. 17 2 GeneralizationofChebyshevNodes.......................... 21 3 LebesgueFunctions ....................................... 23 4 PropertiesofPairsofAuxiliaryFunctions..................... 30 5 OptimalNodesforLagrangePolynomialInterpolation .......... 32 References..................................................... 35 Green’sFunctions:Taking AnotherLookatKernelApproximation, RadialBasisFunctions,andSplines ................................ 37 GregoryE. Fasshauer 1 Introduction.............................................. 37 2 TowardanIntuitiveInterpretationofNativeSpaces............. 39 2.1 WhatistheCurrentSituation?....................... 39 2.2 Mercer’sTheoremandEigenvalueProblems........... 41 2.3 Green’sFunctionsandEigenfunctionExpansions....... 42 2.4 GeneralizedSobolevSpaces ........................ 45 3 FlatLimits............................................... 48 3.1 InfinitelySmoothRBFs ............................ 49 vii viii Contents 3.2 FinitelySmoothRBFs ............................. 50 4 StableComputation ....................................... 53 4.1 AnEigenfunctionExpansionforGaussians............ 53 4.2 TheRBF-QRAlgorithm............................ 54 5 DimensionIndependentErrorBounds........................ 57 5.1 TheCurrentSituation .............................. 57 5.2 NewResultson(Minimal)Worst-CaseWeightedL 2 Error ............................................ 58 6 Summary ................................................ 60 References..................................................... 61 Sparse Recovery Algorithms: Sufficient Conditions in Terms ofRestrictedIsometryConstants .................................. 65 SimonFoucart 1 Introduction.............................................. 65 1.1 BasisPursuit ..................................... 66 1.2 IterativeHardThresholding......................... 66 1.3 CompressiveSamplingMatchingPursuit.............. 66 2 RestrictedIsometryConstants............................... 66 3 BasisPursuit ............................................. 69 4 IterativeHardThresholding................................. 72 5 CompressiveSamplingMatchingPursuit ..................... 74 References..................................................... 77 LagrangeInterpolationandNewAsymptoticFormulaefortheRiemann ZetaFunction .................................................. 79 MichaelI.Ganzburg 1 Introduction.............................................. 79 2 LagrangeInterpolation..................................... 80 3 AsymptoticBehavioroftheInterpolationError ................ 83 4 AsymptoticFormulaeforζ(s) .............................. 86 5 L (−1,1)-AsymptoticsandCriteriaforζ(s)=0andζ(s)(cid:2)=0 ... 89 p 6 Remarks................................................. 92 References..................................................... 92 ActiveGeometricWavelets ....................................... 95 ItaiGershtanskyandShaiDekel 1 Introduction.............................................. 95 2 TheoreticalBackground.................................... 98 2.1 A Jackson Estimate for Piecewise Polynomial ApproximationUsingNon-convexDomains........... 98 2.2 AdaptiveLocalSelectionoftheWeightμ .............100 3 OverviewoftheAGWAlgorithm............................103 4 ExperimentalResults ......................................108 References.....................................................109 Contents ix InterpolatingCompositeSystems .................................. 111 PhilippGrohs 1 Introduction..............................................111 2 CompositeDilationSystems ................................112 3 InterpolatingSystems......................................114 4 Shearlets ................................................118 5 Conclusion...............................................120 References.....................................................120 WaveletsandFrameletsWithintheFrameworkofNonhomogeneous WaveletSystems ................................................ 121 BinHan 1 Introduction..............................................121 2 NonhomogeneousWaveletSystemsinL (Rd) .................125 2 3 Frequency-Based Nonhomogeneous Dual Framelets intheDistributionSpace ...................................132 4 WaveletsandFrameletsinFunctionSpaces ...................140 5 WaveletsandFrameletsDerivedfromFilterBanks .............145 6 DiscreteFrameletTransformandItsBasicProperties ...........149 7 DirectionalTightFrameletsinL (Rd)andProjectionMethod....154 2 References.....................................................158 CompactlySupportedShearlets ................................... 163 GittaKutyniok,JakobLemvig,andWang-QLim 1 Introduction..............................................164 1.1 DirectionalRepresentationSystems ..................165 1.2 AnisotropicFeatures, Discrete Shearlet Systems, andQuestforSparseApproximations.................166 1.3 ContinuousShearletSystems........................167 1.4 Applications......................................168 1.5 Outline ..........................................170 2 2DShearlets .............................................171 2.1 Preliminaries .....................................171 2.2 ClassicalConstruction .............................172 2.3 ConstructingCompactlySupportedShearlets ..........173 3 SparseApproximations ....................................175 3.1 Cartoon-likeImageModel..........................175 3.2 OptimallySparseApproximationofCartoon-like Images ..........................................176 4 Shearletsin3DandBeyond ................................177 4.1 Pyramid-AdaptedShearletSystems ..................178 4.2 SparseApproximationsof3DData...................181 5 Conclusions..............................................184 References.....................................................184 x Contents ShearletsonBoundedDomains.................................... 187 GittaKutyniokandWang-QLim 1 Introduction..............................................187 1.1 OptimallySparseApproximationsofCartoon-like Images ..........................................188 1.2 ShortcomingsofthisCartoon-likeModelClass ........189 1.3 OurModelforCartoon-likeImagesonBounded Domains.........................................189 1.4 ReviewofShearlets................................191 1.5 SurprisingResult..................................191 1.6 MainContributions................................192 1.7 Outline ..........................................193 2 CompactlySupportedShearlets .............................193 2.1 CompactlySupportedShearletFramesforL2(R2) ......193 2.2 CompactlySupportedShearletFramesforL2(Ω ).......195 3 OptimalSparsityofShearletsonBoundedDomains ............196 3.1 MainTheorem1 ..................................196 3.2 ArchitectureoftheProofofTheorem1 ...............197 4 ProofofTheorem1 .......................................198 4.1 Case1:TheSmoothPart ...........................198 4.2 Case2:TheNon-SmoothPart .......................199 5 Discussion ...............................................205 References.....................................................206 On Christoffel Functions and Related Quantities for Compactly SupportedMeasures............................................. 207 D.S.Lubinsky 1 Introduction..............................................207 2 ProofofTheorem6 .......................................214 References.....................................................219 Exact Solutions of Some Extremal Problems ofApproximationTheory......................................... 221 A.L.Lukashov 1 Introduction..............................................221 2 Proofs...................................................223 References.....................................................228 A LagrangeInterpolationMethodby Trivariate CubicC1 Splines ofLowLocality ................................................. 231 G.Nu¨rnbergerandG.Schneider 1 Introduction..............................................231 2 Preliminaries.............................................232 3 AUniformPartitionConsistingofTetrahedraandOctahedra.....235 4 A(Partial)Worsey–FarinRefinementof♦ ....................239