New Trends in Approximation Theory

277 Pages·2018·5.53 MB·English

by  Javad Mashreghi

Fields Institute Communications 81
The Fields Institute for Research in Mathematical Sciences
Javad Mashreghi
Myrto Manolaki
Paul Gauthier Editors
New Trends in
Approximation
Theory
In Memory of André Boivin
Fields Institute Communications
Volume 81
FieldsInstituteEditorialBoard:
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HuaxiongHuang,DeputyDirectoroftheInstitute
JamesG.Arthur,UniversityofToronto
KennethR.Davidson,UniversityofWaterloo
LisaJeffrey,UniversityofToronto
BarbaraLeeKeyfitz,OhioStateUniversity
ThomasS.Salisbury,YorkUniversity
JurisSteprans,YorkUniversity
NorikoYui,Queen’sUniversity
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Javad Mashreghi • Myrto Manolaki • Paul Gauthier
Editors
New Trends in
Approximation Theory
In Memory of André Boivin
123
Editors
JavadMashreghi MyrtoManolaki
DépartementdeMathématiquesetde DepartmentofMathematicsandStatistics
Statistique UniversityofSouthFlorida
UniversitéLaval Tampa,FL,USA
Québec,QC,Canada
PaulGauthier
DépartementdeMathématiquesetde
Statistique
UniversitédeMontréal
Montréal,QC,Canada
ISSN1069-5265 ISSN2194-1564 (electronic)
FieldsInstituteCommunications
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MathematicsSubjectClassification(2010):30E10,30H05,32A35,32E30,41A20
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Preface
The international conference entitled “New Trends in Approximation Theory”
was held at the Fields Institute, in Toronto, from July 25 to July 29, 2016. The
conference (which received financial support from the Fields Institute and CRM)
was fondly dedicated to the memory of our unique friend and colleague André
Boivin,whogavetirelessserviceinCanadauntiltheverylastmomentofhislifein
October 2014. The impact of his warm personality and his fine work on Complex
ApproximationTheorywasreflectedbythemathematicalexcellenceandthewide
researchrangeofthe37participants.Intotaltherewere27talks,deliveredbywell-
establishedmathematiciansandyoungresearchers.Inparticular,19invitedlectures
were delivered by leading experts of the field, from 8 different countries (USA,
France,Canada,Ireland,Greece,Spain,Israel,Germany).Videosandslidesofthe
presentationscanbefoundatthefollowinglink:
https://www.fields.utoronto.ca/video-archive/event/1996
The wide variety of presentations composed a mosaic of multiple aspects of
Approximation Theory and highlighted interesting connections with important
contemporary areas of analysis. In particular, the main topics that were discussed
includethefollowing:
1. ApplicationsofApproximationTheory(isoperimetricinequalities,construction
ofentireorder-isomorphisms,dynamicalsampling);
2. Approximationbyharmonicandholomorphicfunctions(andespeciallyuniform
andtangentialapproximation);
3. Polynomialandrationalapproximation;
4. Zerosofapproximantsandzero-freeapproximation;
5. Tools used in Approximation Theory (analytic capacities, Fourier and Markov
inequalities);
6. Approximationoncomplexmanifolds(Riemannsurfaces),andapproximationin
productdomains;
7. Approximationinfunctionspaces(HardyandBergmanspaces,discalgebra,de
Branges–Rovnyakspaces);
8. BoundarybehaviouranduniversalitypropertiesofTaylorandDirichletseries.
v
vi Preface
Throughout the conference there was a very creative and friendly atmosphere,
withmanyinterestingdiscussionsandmathematicalinteractionswhich,hopefully,
will lead to future collaborations. The last talks of the conference were devoted
to the main contributions of André Boivin in Approximation Theory and his
collaborationswhicharepresentedinthefirstchapterinfurtherdetail.
Montréal,QC,Canada PaulGauthier
Tampa,FL,USA MyrtoManolaki
Québec,QC,Canada JavadMashreghi
A Riemann surface, full of some (among many) good friends of André Boivin, during the
conference “New Trends in Approximation Theory” which was held in his memory (Fields
Institute,July2016).
Contents
TheLifeandWorkofAndréBoivin ........................................... 1
PaulGauthier,MyrtoManolakiandJavadMashreghi
ANoteontheDensityofRationalFunctionsinA1.˝/..................... 27
JavierFalcó,VassiliNestoridis,andIliasZadik
Approximation by Entire Functions in the Construction
ofOrder-IsomorphismsandLargeCross-Sections........................... 37
MaximR.Burke
ApproximationbySolutionsofEllipticEquationsandExtension
ofSubharmonicFunctions...................................................... 71
PaulGauthierandPetrV.Paramonov
ApproximationintheClosedUnitBall........................................ 89
JavadMashreghiandThomasRansford
AThoughtonApproximationbyBi-AnalyticFunctions.................... 131
DmitryKhavinson
ChebyshevPolynomialsAssociatedwithaSystemofContinua ............ 137
IsaacDeFrain
ConstrainedL2-ApproximationbyPolynomialsonSubsets
oftheCircle....................................................................... 151
LaurentBaratchart,JulietteLeblond,andFabienSeyfert
ExtremalBoundsofTeichmüller-Wittich-Belinski˘ıTypeforPlanar
QuasiregularMappings ......................................................... 173
AnatolyGolberg
FamiliesofUniversalTaylorSeriesDependingonaParameter............ 201
EvgenyAbakumov,JürgenMüller,andVassiliNestoridis
InterpolationbyBoundedAnalyticFunctionsandRelatedQuestions..... 215
ArthurA.Danielyan
ix
x Contents
OnTwoInterpolationFormulasforComplexPolynomials ................. 225
RichardFournierandStephanRuscheweyh
OperatorswithSimpleOrbitalBehavior...................................... 235
GabrielT.Praˇjituraˇ
TaylorSeries,UniversalityandPotentialTheory ............................ 247
StephenJ.Gardiner
SubharmonicImagesofaConvergentSequence............................. 265
PaulGauthierandMyrtoManolaki