Operator-Related Function Theory and Time-Frequency Analysis: The Abel Symposium 2012

204 Pages·2015·3.07 MB·English

by  Karlheinz Gröchenig

Abel Symposia 9
Karlheinz Gröchenig
Yurii Lyubarskii
Kristian Seip E ditors
Operator-Related
Function Theory
and Time-Frequency
Analysis
The Abel Symposium 2012
ABEL SYMPOSIA
Edited by the Norwegian Mathematical Society
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Karlheinz GroRchenig • Yurii Lyubarskii (cid:129)
Kristian Seip
Editors
Operator-Related
Function Theory and
Time-Frequency Analysis
The Abel Symposium 2012
123
Editors
KarlheinzGroRchenig YuriiLyubarskii
FacultyofMathematics KristianSeip
UniversityofVienna DepartmentofMathematicalSciences
Vienna NorwegianUniversityofScience
Austria andTechnology
Trondheim
Norway
ISSN2193-2808 AbelSymposia
ISBN978-3-319-08556-2 ISBN978-3-319-08557-9(eBook)
DOI10.1007/978-3-319-08557-9
SpringerChamHeidelbergNewYorkDordrechtLondon
LibraryofCongressControlNumber:2014956015
MathematicsSubjectClassification(2010):35-XX,42-XX,47-XX,94-XX
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Foreword
The Niels Henrik Abel Memorial Fund was established by the Norwegian gov-
ernmenton January 1, 2002. The main objective is to honor the great Norwegian
mathematicianNielsHenrikAbelbyawardinganinternationalprizeforoutstanding
scientificworkinthefieldofmathematics.Theprizeshallcontributetowardsraising
the status of mathematics in society and stimulate the interest for science among
schoolchildrenandstudents.Inkeepingwith thisobjectivetheBoardof theAbel
FundhasdecidedtofinanceanannualAbelSymposium.Thetopicmaybeselected
broadly in the area of pure and applied mathematics. The Symposia should be
at the highest international level, and serve to build bridges between the national
and international research communities. The Norwegian Mathematical Society is
responsiblefortheevents.Ithasalsobeendecidedthatthecontributionsfromthese
Symposiashouldbe presentedin a seriesofproceedings,andSpringerVerlaghas
enthusiastically agreed to publish the series. The board of the Niels Henrik Abel
Memorial Fund is confident that the series will be a valuable contribution to the
mathematicalliterature.
HelgeHolden
ChairmanoftheboardoftheNielsHenrikAbelMemorialFund
v

Preface
The topic of the 2012 Abel Symposium was Operator-Related Function Theory
and Time-Frequency Analysis. The symposium centered on these two important
fields of modernmathematicalanalysisand the profoundinterplaybetween them.
Contemporary complex analysis is a powerful tool in harmonic and functional
analysis, probability theory, and in applied areas such as control theory and
informationtheory.Time-frequencyanalysisoriginatedwithinquantummechanics
and signal analysis and has since grown into an independent mathematical dis-
cipline, embracing different areas from harmonic analysis and combinatorial and
geometricalanalysis.Methods,approaches,and–perhapsevenmoreimportantly–
the philosophy of time-frequency analysis allow us to reexamine known results,
discover new unexplored areas in classical function theory, and also to establish
surprisingandprofoundrelationsbetweenproblemsarisinginseeminglydisparate
areas of mathematics. The purpose of these Abel Symposium proceedings is to
presentaselectionofthelatestexcitingresultsbytheworld’sleadingresearchersin
thesetwoareasofresearch.
The Abel Symposium was hosted at the Norwegian Academy of Science
and Letters, Oslo, August 20–24, 2012. Attendance was by invitation only,
and the symposium had a total of 61 participants, 23 of whom were from
Norwegianuniversities.TheScientificCommitteeconsistedofKarlheinzGröchenig
(Vienna),MichaelLacey(GeorgiaTech),JoaquimOrtega-Cerdà(Barcelona),Yurii
Lyubarskii(Trondheim),KristianSeip(Trondheim)andMikhailSodin(TelAviv).
Talks were presentedby Artur Avila (Paris), IngridDaubechies(Duke),László
Erdo˝s(Munich),HansFeichtinger(Vienna),StéphaneJaffard(Paris-Est), Izabella
Łaba (Vancouver), Nikolai Makarov (Caltech), Clément Mouhot (Cambridge),
Alexander Olevskii (Tel Aviv), Alexei Poltoratskii (Texas A&M), Eero Saksman
(Helsinki), Eric Sawyer (Hamilton), Johannes Sjöstrand (Dijon), Sergei Treil
(Providence)andYosefYomdin(Rehovot).
vii
viii Preface
These proceedingsinclude selected lecturespresented at the Abel Symposium.
Theeditorswouldliketothankallspeakerswhohavepresentedtheirtalksinthese
proceedings.
The participants were: Aleksei Aleksandrov, Alexandru Aleman, Artur Avila,
Anton Baranov, Yurii Belov, Christian Berg, Bo Berndtsson, Toke Meier Carlsen,
Dmitrii Chelkak, Ole Christensen, Jacob S. Christiansen, Ingrid Daubechies,
Trond Digernes, Mats Ehrnström, Kjersti Solberg Eikrem, László Erdo˝s, Hans
Feichtinger,JohnErikFornæss,SigridGrepstad,KarlheinzGröchenig,AnttiHaimi,
HaraldHanche-Olsen,HåkanHedenmalm,HelgeHolden,AlexanderIgamberdiev,
Marius Irgens, Georgy Ivanov, Stéphane Jaffard, Kenneth H. Karlsen, Izabella
Łaba, Michael Lacey, Magnus Landstad, Peter Lindqvist, Alexander Logunov,
Yurii Lyubarskii, Erik Løw, Nikolai Makarov, Eugenia Malinnikova, Clément
Mouhot, Alexander Olevskii, Jan-Fredrik Olsen, Eduard Ortega, Joaquim Ortega-
Cerdà,HenrikLaurbergPedersen,Karl-MikaelPerfekt,AlexeiPoltoratskii,Sandra
Pott, Maria Carmen Reguera, Nils Henrik Risebro, Eero Saksman, Eric Sawyer,
Kristian Seip, Sigmund Selberg, Johannes Sjöstrand, Arne Stray, Alexey Tochin,
SergeiTreil,FranciscoVillarroya,ErlendFornæssWold,YosefYomdinandPavel
Zatitskyi.
We gratefully acknowledge the financial support providedby the Niels Henrik
Abel Memorial Fund, and by the Centre for Advanced Study at the Norwegian
Academyof Science and Letters. The kind assistance of Tanja Opheim and Marit
F.Strømisgreatlyappreciated.Lastly,ourthanksgotoOleFredrikBrevigforhis
assistanceinthepreparationoftheseproceedings.
Trondheim,Norway YuriiLyubarskii
KristianSeip
Vienna,Austria KarlheinzGröchenig
Contents
A BridgeBetweenGeometricMeasureTheoryandSignal
Processing:MultifractalAnalysis.............................................. 1
P.Abry,S.Jaffard,andH.Wendt
1 Introduction.................................................................... 1
2 PointwiseRegularity:TwoExamples........................................ 7
2.1 TaylorPolynomialandPeanoDerivatives............................. 7
2.2 LévyFunctions......................................................... 8
2.3 BinomialMeasures...................................................... 12
3 MathematicalNotionsPertinentinMultifractalAnalysis................... 15
3.1 ToolsDerivedfromGeometricMeasureTheory...................... 15
3.2 ToolsDerivedfromPhysicsandSignalProcessing................... 19
4 WaveletBasedScalingFunctions ............................................ 21
4.1 WaveletBases........................................................... 21
4.2 TheWaveletScalingFunction.......................................... 25
4.3 WaveletLeaders......................................................... 30
4.4 Estimationofthep-Oscillationandp-Variation...................... 32
5 TheCurseofConcavity....................................................... 34
5.1 MathematicalExamplesofNon-concaveSpectra..................... 35
5.2 TheLargeDeviationLeaderSpectrum ................................ 39
5.3 TheQuantileLeaderSpectrum......................................... 42
5.4 WeightedLegendreTransform......................................... 46
6 MultifractalAnalysisofNon-locallyBoundedFunctions................... 47
6.1 ConvergenceandDivergenceRatesforWaveletSeries............... 47
6.2 PointwiseLq Regularity:TheUseofq-Leaders ...................... 50
References......................................................................... 54
LocalandGlobalGeometryofPronySystemsandFourier
ReconstructionofPiecewise-SmoothFunctions .............................. 57
D.BatenkovandY.Yomdin
1 Introduction.................................................................... 57
2 ThePronyProblem............................................................ 58
ix