Approximation by Solutions of Partial Differential Equations

206 Pages·1992·15.07 MB·English

by  D. H. Armitage (auth.)

Approximation by Solutions
of Partial Differential Equations
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SeriesC:MathematicalandPhysicalSciences-Vol.365
Approximation by Solutions
of Partial Differential Equations
edited by
B. Fuglede
Department of Mathematics,
University of Copanhagen,
Copenhagen, Denmark
M. Goldstein
Departmem of Mathematics,
Arizona State University,
Tempa, Arizona, U.SA
W. Haussmann
Department of Mathematics,
University of Duisburg,
Duisburg, Germany
W. K. Hayman
Department of Mathematics,
University of York,
Heslington, York, U.K.
and
L.Rogge
Department of Mathematics,
University of Duisburg,
Duisburg, Germany
Springer Science+Business Media, B.v.
Proceedings of the NATO Advanced Research Workshop on
Approximation by Solutions of Partial Differential Equations, Quadrature Formulae,
and Related Topics
Hanstholm, Denmark
July 8-12, 1991
ISBN 978-94-010-5074-6 ISBN 978-94-011-2436-2 (eBook)
DOI 10.1007/978-94-011-2436-2
Printed an acid-tree paper
AII Rights Reserved
© 1992 Springer Science+Business Media Dardrecht
Originally published by Kluwer Academic Publishers in 1992
Saftcaver reprint of the hardcaver 1s t editian 1992
No part of the material protected by this copyright notice may be reproduced or
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permission from the copyright owner.
TABLE OF CONTENTS
Participants vii
ScientificProgramofthe Conference ix
Preface xi
CharacterizationsofBallsand Stripsvia Harmonic Quadrature
D. H. Armitageand M. Goldstein 1
Better than Uniform Approximationon Closed Setsby Harmonic
Functionswith SingularitiesandApplications
D. H. Armitageand M. Goldstein 11
Uniform Approximation byGlobal HarmonicFunctions
T. Bagby andP. M. Gauthier 15
Weighted IJ' Approximation by HolomorphicFunctions
A. Boivin, J. Mateu and J. Verdera 27
Boundary CorrespondenceofUnivalent Harmonic Mappings
from the Unit Discontoa Jordan Domain
D. Bshouty and W. Hengartner 51
Finely Open Setsin the Limit Set ofa Finetely Generated
Kleinian Group
A. E. EremenkoandT. J. Lyons 61
Rational Approximation tothe FresnelIntegral
W. H. J. Fuchs and W. K. Hayman 69
Axiomatic Approximation
P. M. Gauthier, J. Heinonen and D. Zwick 79
CharacterizationofOpen Stripsby Harmonic Quadrature
M. Goldstein, W. Haussmann and1. Rogge 87
Mean Value Theorems and Best LI-Approximation
W. Haussmann, L. Wehrend and K. Zeller 93
Mapping Propertiesofthe Wave Operatorsin Scattering Theory
A. Jensen 103
vi TABLEOFCONTENTS
Harmonicity Modulus andApplications to theApproximation
by Polyharmonic Functions
O. I. Kounchev III
A ComparisonofHarmonicand Maximal Measuresfor
Rational Functions
M. Lyubich and A. Volberg 127
The Role ofthe Hilbert Transformin 2-DimensionalAerodynamics
T. Murai 141
Approximation by HarmonicFunctions and the Dirichlet Problem
1. Netuka 155
On the Existence-Problemfor Gaufi-Quadratureon the Sphere
M. Reimer 169
A Bibliographic Surveyofthe Pompeiu Problem
L. Zalcman 185
Problems 195
LIST OF PARTICIPANTS
David Armitage, Department of Pure Mathematics, Queen's University,
Belfast BT7INN, Northern Ireland
Thomas Bagby, Department ofMathematics, Indiana University, Swain
Hall-East, Bloomington,IN47405, U.S.A.
MonikaBenninghoff-Grasnick, KaslinerStr. 13,D-5064 Rasrath, Germany
Andre Boivin, DepartmentofMathematics, University ofWestern Ontario,
London, Ontario, Canada N6A 5B7
Alex Eremenko, Department ofMathematics, Purdue University, West
Lafayette,Indiana, U.S.A.
BentFuglede,MathematicsInstitute,Universitetsparken5,DK-2100Copen
hagen, Denmark
PaulM.Gauthier,DepartmentdeMathematiquesetdeStatistique, Univer
sitedeMontreal, Montreal, Quebec, CanadaH3C3J7
Myron Goldstein, Department ofMathematics, Arizona State University,
Tempe, AZ85287, U.S.A.
Uwe Grasnick, Nordstern Lebensversicherungs AG, Konrad-Adenauer
Ufer 23, D-5000Kaln 1, Germany
Werner Haussmann, Fachbereich Mathematik, Universitiit Duisburg,
Lotharstr. 65, D-4100Duisburg, Germany
Walter K. Hayman, DepartmentofMathematics, UniversityofYork, Hes
lington, York YOI 5DD, England
Walter Hengartner, DepartmentdeMathematiqueset deStatistique, Uni
versiteLaval, Cite Universitaire, Quebec, Canada GIK 7P4
Arne.Jensen, DepartmentofMathematicsandComputerScience,Institute
for Electronic Systems, Aalborg University, Fredric Bajers Vej 7,
DK-9220Aalborg, Denmark
Ognyan Kounchev, Institute ofMathematics, Bulgarian Academy ofSci
ences, Acad. G. BonchevStr. 8, BG-I113Sofia, Bulgaria
vii
viii LISTOFPARTICIPANTS
Peter Kuhach, Fachbereich Mathematik, Universitiit Dortmund, Post
fach 500500, D-4600Dortmund50
John Lewis, Mathematics Department, UniversityofKentucky, Lexington,
KY40506-0027, U.S.A.
Terry J. Lyons, Department of Mathematics, University of Edinburgh,
James Clerk Maxwell Building, TheKing's Buildings, Edinburgh,
EH93JZ, Scotland
Joan Mateu Bennassar, Departament de Matematica Aplicada I,
UniversitatPolitechnicadeCatalunya,Diagonal647,E-08028Bar
celona, Spain
Stefanie Messal, Fachbereich Mathematik, Universitiit Duisburg, Lothar
str. 65, D-4100Duisburg, Germany
Takafumi Murai, Department ofMathematics, School ofScience, Nagoya
University, Chikusa-ku, Nagoya464-01, Japan
Ivan Netuka, Matematicky Ustav Univerzity Karlovy, Sokolovski 83,
CZ-186 00Praha, Czechoslovakia
Joan Orobitg, Departament de Matematica AplicadaII, Pau Gargallo 5,
E-08028Barcelona, Spain
WellingtonH.Ow, DepartmentofMathematics, MichiganStateUniversity,
East Lansing, MI48824, U.S.A.
MichaelPedersen,MathematicalInstitute,Technical UniversityofDenmark,
Building303, DK-2800Lyngby, Denmark
Manfred Reimer, Fachbereich Mathematik, Universitiit Dortmund, Post
fach 500500, D-4600Dortmund50, Germany
Lothar Rogge, Fachbereich Mathematik, Universitiit Duisburg, Lothar
str. 65, D-4100Duisburg, Germany
TordSjodin,DepartmentofMathematics, Umea University,S-90187Umea,
Sweden
JoachimStockIer, Fachbereich Mathematik, Universitiit Duisburg, Lothar
str. 65, D-4100Duisburg, Germany
Joan Verdera, Departament de Matematiques, Universitat Autonoma de
Barcelona, E-08193Bellaterra, Barcelona, Spain
Lothar Wehrend, Honigsberger Str. 33, D-4330Miilheim, Germany
Lawrence Zalcman, Department of Mathematics, Bar-Ilan University,
52900Ramat-Gan, Israel
DanielZwick,DepartmentofMathematics,UniversityofVermont,16Colch
esterAve., Burlington, VT 05405, U.S.A.
SCIENTIFIC PROGRAM OF THE CONFERENCE
Monday, July 8:
Morning Session: Chairman M. Goldstein
9:00 OpeningoftheWorkshop
9:10- 10:10 W. K. Hayman: RationalApproximation tothe Fresnel
Integral
10:30- 11:30 T. Bagby: Uniform Approximation by Holomorphicand
HarmonicFunctionson Closed SubsetsofManifolds
11:30- 12.30 P. M. Gauthier: Uniform ApproximationofUnbounded
Sets:Applications
Afternoon Session: Chairman W. K. Hayman
15:00- 16:00 D. Zwick: Best Uniform Approximation by Solutions
andSupersolutionsofElliptic Equations
16:30- 17:30 O.1. Kounchev: Direct andInverseTheoremsfor the
Approximation Through Polyharmonicand Polyanalytic
Functions
17:30- 18:30 W. Haussmann: Mean Values and Best L1-Approximation
Tuesday, July 9:
Morning Session: Chairman J. Lewis
9:00- 10:00 L. Zalcman: Recent Reults on Pompeiu Problems
10:30- 11.30 L. Rogge: CharacterizationofOpen StripsviaHarmonic
Quadrature
11:30- 12:30 D. Armitage: CharacterizationsofBalls and Stripsvia
Harmonic Quadrature
Afternoon Session: Chairman T. Murai
15:00- 16:00 A. Eremenko: DistributionofValuesofMeromorphic
Functionsfrom the PointofViewofPotentialTheory
16:30- 17:30 1. Netuka: Approximation by HarmonicFuntions and
the Dirichlet Problem
17:30- 18:30 M. Reimer: Onthe Existence ProblemofGauf\
Quadratures
ix
x SCIENTIFICPROGRAMOFTHECONFERENCE
Wednesday, July 10:
Morning Session: Chairman B. Fuglede
9:00- 10:00 J. Lewis: On Pseudospheres
10:30- 11:30 T. J. Lyons: Stratonovich Stochastic Calculus Bounds
on, and the TailsofHeat Kernels
11:30- 12:30 W. Hengartner: Boundary CorrespondenceofUnivalent
HarmonicMappings
Afternoon Session: Chairman T. Lyons
15:00- 16:00 J. Verdera: The Uniform Approximation Problemfor
the Square ofthe Cauchy Riemann Operator and a
Weak TypeInequalityfor Analytic Capacity
16:30- 17:30 M. Goldstein: Betterthan Uniform Approximationon
Closed Setsby Harmonic Functionswith Singularities
andApplications
17:30- 18:30 W. H. Ow: A CharacterizationofHarmonic Arakelyan
Sets
Evening Session: Chairman M. Goldstein
20.00- 21.30 Problem Session
Thursday, July 11:
Morning Session: Chairman L. Zalcman
9:00- 10:00 T. Murai:The Roleofthe Hilbert Transformin
2-DimensionalAerodynamics
10:30- 11:30 M. Pedersen: Regularity ofPointwise Boundary
Control Systems
11:30- 12:30 A. Jensen: Mapping PropertiesoftheWave Operator
in Scattering Theory