Approximation Theory XIII: San Antonio 2010

419 Pages·2012·4.321 MB·English

by  B. A. Bailey (auth.), Marian Neamtu, Larry Schumaker (eds.)

Springer Proceedings in Mathematics
Volume 13
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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
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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