Harmonic analysis and rational approximation: their roles in signals, control and dynamical systems

307 Pages·2006·10.13 MB·English

by  Jean-Daniel Fournier

Lecture Notes
in Control and Information Sciences 327
Editors:M.Thoma · M.Morari
J.-D. Fournier (cid:1) J. Grimm (cid:1) J. Leblond
J. R. Partington (Eds.)
Harmonic Analysis
and Rational
Approximation
Their Ro^les in Signals, Control
and Dynamical Systems
With47Figures
SeriesAdvisoryBoard
F.Allgo¨wer·P.Fleming·P.Kokotovic·A.B.Kurzhanski·
H.Kwakernaak·A.Rantzer·J.N.Tsitsiklis
Editors
Dr.J.-D.Fournier Prof.J.R.Partington
Dr.J.Grimm
UniversityofLeeds
Dr.J.Leblond
SchoolofMathematics
De´partementARTEMIS LS29JTLeeds
CNRSandObservatoiredelaCoˆted’Azur UnitedKingdom
BP4229
06304NiceCedex4
France
ISSN0170-8643
ISBN-10 3-540-30922-5 SpringerBerlinHeidelbergNewYork
ISBN-13 978-3-540-30922-2 SpringerBerlinHeidelbergNewYork
LibraryofCongressControlNumber:2005937084
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In memoriam Macieja Pindora
Preface
ThisbookisanoutgrowthofasummerschoolthattookplaceontheIslandof
Porquerolles in September 2003. The goal of the school was mainly to teach
certain pieces of mathematics to practitioners coming from three different
communities: signal, control and dynamical systems theory. Our impression
was indeed that, in spite of their great potential applicability, 20th century
developmentsinapproximationtheoryandFouriertheory,whilecommonplace
among mathematicians, are unknown or under-appreciated within the above-
mentioned communities. Specifically, we had in mind:
• someadvancesinanalytic,meromorphicandrationalapproximationtheory,
as well as their links with identification, robust control and stabilization
of infinite-dimensional systems;
• the rich correspondences between the complex and real asymptotic be-
havior of a function and its Fourier transform, as already described, for
instance, in Wiener’s books.
In this respect, it is noticeable that in the last twenty years, much effort has
been devoted to the research and teaching of recent decomposition tools, like
wavelets or splines, linked to real analysis. From the early stages, we shared
the view that, in contrast, research in certain fields suffers from the lack of a
working knowledge of modern Fourier analysis and modern complex analysis.
Finally, we felt the need to introduce at the core of the school a proba-
bilistic counterpart to some of the questions raised above. Although familiar
to specialists of signal and dynamical systems theory, probability is often ig-
noredbymembersofthecontrolandapproximationtheorycommunities.Yet
we hope to convey to the reader the conviction that there is room for fa-
scinating phenomena and useful results to be discovered at the junction of
probability and complex analysis.
This book is not just a proceedings of the summer school, since the con-
tributions made by the speakers have been totally rewritten, anonymously
refereed and edited in order to reflect some of the common themes in which
the authors are interested, as well as the diversity of the applications. The
VIII Preface
contributors were asked to imagine addressing a fellow-scientist with a non-
negligible but modest background in mathematics.
In drawing the boundaries between the chapters of the book, we have also
triedtoeliminateredundancy,whileallowingforrepetitionofathemeasseen
from different points of view.
We begin in Part I with a general introduction from the late Maciej Pin-
dor. He surveys the conceptual and practical value of complex analyticity,
both in the physical and the conjugate Fourier variables, for physical theories
originally built in the real domain. Obstacles to analytic extension, like polar
singularities known as “resonances”, a key concept of the school, turn out to
have themselves a physical meaning. It is illustrated here by means of optical
dispersion relations and the scattering of particles.
Part II of this book contains basic material on the complex analysis and
harmonicanalysisunderlyingthefurtherdevelopmentspresentedinthebook.
Candelpergherwritesoncomplexanalysis,inparticularanalyticcontinuation
and the use of Borel summability and Gevrey series. Partington gives an
account of basic harmonic analysis, including Fourier, Laplace and Mellin
transforms, and their links with complex analysis.
Part III contains further basic material, explaining some of the aspects
of approximation theory. Pindor presents the theory of Pad´e approximation,
including convergence issues. Levin and Saff explain how potential theoretic
tools such as capacity play a role in the study of efficient polynomial and
rationalapproximation,andanalysesomeweightedproblems.Partingtondis-
cussestheuseofbasesofrationalfunctions,includingorthogonalpolynomials,
Szeg¨o bases, and wavelets.
FinallyPartIVcompletesthefoundationsbyatourinprobabilitytheory.
The driving force behind the order emerging from randomness, the central
limittheorem,isexplainedbyCollet,includingconvergenceandfractalissues.
Dujardin gives an account of the properties of random real polynomials, with
particularreferencetothedistributionoftheirrealandcomplexroots.Pindor
puts rational approximation into a stochastic context, the basic idea being to
obtain rational interpolants to noisy data.
Themajorapplicationofthethemesofthisbookliesinsignalandcontrol
theory, which is treated in Part V. Deistler gives a thorough treatment of
the spectral theory of stationary processes, leading to an account of ARMA
and state space systems. Cuoco’s paper treats the power spectral density of
physical systems and its estimation, to be used in the extraction of signals
out of noisy data. Olivi continues some of the ideas of Parts II and III, and,
underthegeneralumbrellaoftheLaplacetransformincontroltheory,discus-
ses linear time-invariant systems, controllability and rational approximation.
Baratchart uses Laplace–Fourier transform techniques in giving an account
of recent work analysing problems originating in the identification of linear
systems subject to perturbations. In a final return to the perspective of the
Introduction, Parts VI and VII shows the roˆle of the previously-discussed
toolsinextremelydiversedomainsofphysics.InPartVI,somemathematical
Preface IX
aspects of dynamical systems theory are discussed. Biasco and Celletti are
concernedwithcelestialmechanicsandtheuseofperturbationtheorytoana-
lyseintegrableandnearly-integrablesystems.Baladigivesabriefintroduction
toresonancesinhyperbolicandhamiltoniansystems,consideredviathespec-
traofcertaintransferoperators.PartVIIisdevotedtoamodernapproachto
two classical physics problems. Borgnat is concerned with turbulence in fluid
flow; he discusses which tools, including the Mellin transform, can be adap-
ted to reveal the various statistical properties of intermittent signals. Finally,
Bondu and Vinet give an account of the high-performance control and noise
analysis required at the gravitational waves VIRGO antenna.
Last but not least, our thanks go to the authors of the 17 contributions
gathered in this book, as well as to all those who have helped us produce it,
with particular mention of the anonymous referees.
Nice (France), Sophia-Antipolis (France), Leeds (U.K.), July 2005.
The editors: Jean-Daniel Fournier,
Jos´e Grimm,
Juliette Leblond,
Jonathan R. Partington.
X Preface
Maciej PINDOR
Our colleague Dr. Maciej Pindor of Poland, the friend, collaborator and vi-
sitor of Jean-Daniel Fournier (JDF), died on Saturday 5th July 2003 at the
Nice Observatory. Apparently, he was on his way to work from the “Pavil-
lon Magn´etique”, where he was staying, to his office at “CION”. His death
was attributed to cardiac problems. He was 62 years old. Some colleagues
were present, including the Director of the “Observatoire de la Coˆte d’Azur”
(OCA) and JDF, when help arrived.
MaciejPindorwasaseniorlecturerattheInstituteofTheoreticalPhysics
at the University of Warsaw. He performed his research work with the same
care that he devoted to his teaching duties. He was a specialist in complex
analysis,appliedtosomequestionsoftheoreticalphysics,and,inrecentyears,
to the processing of data; he produced theoretical and numerical solutions,
whichinthisregardshowedaningenuityandreliabilitythatishardtomatch.
He taught effective computational methods to young physicists. From the
beginning of the thesis that Ben´edicte Dujardin has been writing under the
direction of JDF, M. Pindor participated in her supervision.
ThecollaborationofJDFandhiscolleagueswithM.Pindorbeganin1996.
Over the years, it was supported by regular or exceptional funding from the
Cassini Laboratory, the Theoretical Physics Institute of Warzaw, the Polish
Academy of Sciences and from OCA (with an associated post in astronomy).
Thus M. Pindor came to Nice several times, and many people knew him. His
genuine modesty made him a very accessible person, and dealings with him
were agreeable and fruitful in all cases.
ForthesummerschoolofPorquerolles,hehadagreedtogivethreecourses,
on three different subjects. In this he was motivated by friendship, scientific
interest, and his acute awareness of the teaching responsibility borne by uni-
versity staff; since then he had overcome the anxiety that he felt towards the
idea of presenting mathematics in front of professional mathematicians. In
particular, he was due to give the opening course, showing the link between
physics and mathematics, treating the ideas of analyticity and resonance. He
produced his notes for the course in good time, and these are therefore in-
cluded under his name in this book and listed in the table of contents. At
Porquerolles his courses were given by three different people. As co-worker
JDF took the topic “rational approximation and noise”. We sincerely thank
the two others: G. Turchetti, himself an old friend of M. Pindor, agreed to
expound the roˆle of analytic continuation and Pad´e approximants in theore-
tical and mathematical physics; E. B. Saff kindly offered to lecture on the
mathematics behind Pad´e approximants.
This book is dedicated to the memory of Maciej Pindor.
This obituary and M. Pindor’s photograph have been included here by
agreement with his widow, Dr. Krystyna Pindor-Rakoczy.
Preface XI
Memories of the Porquerolles School, a word from the
co-directors
As already mentioned in the Preface, we organized the editing of the present
book as a separate scientific undertaking, distinct from the school itself and
with a wider team including J. Grimm and J.R. Partington. Nevertheless we
feel bound to stress that the book is in part the result of the intellectual
and congenial atmosphere created in Porquerolles in September 2003 by the
speakers and the participants. Such moments are to be cherished, and have
rewarded us for our own preparatory work. This seems a natural place to
thank those of our colleagues who contributed to the running of the school,
either as scientists or assistants, including those whose names do not appear
here. Conversely we thank especially Elena Cuoco, who agreed to write a
chapter for the book, although she had not been able to attend the school for
personal reasons.
List of participants
D. Avanessoff (INRIA, Sophia-Antipolis [SA]),
V. Baladi (CNRS, Univ. Jussieu, Paris),
L. Baratchart (INRIA, SA),
L. Biasco (Univ. Rome III, It.),
B. Beckermann (Univ. Lille),
F. Bondu (CNRS, Observatoire de la Coˆte d’Azur [OCA], Nice),
XII Preface
P. Borgnat (CNRS, ENS Lyon),
V. Buchin (Russian Academy of Sciences, Moscow, Russia),
B. Candelpergher (Univ. Nice, Sophia-Antipolis [UNSA]),
A. Celletti (Univ. Rome Tor Vergata, It.),
A. Chevreuil (Univ. Marne la Vall´ee),
C. Cichowlas (ENS Ulm, Paris),
P. Collet (CNRS, Ecole Polytechnique, Palaiseau),
D. Coulot (OCA, Grasse),
F. Deleflie (OCA, Grasse),
M. Deistler (Univ. Tech. Vienne, Aut.),
B. Dujardin (OCA, Nice),
Y. Elskens (CNRS, Univ. Provence, Marseille),
J.-D. Fournier (CNRS, OCA, Nice),
V. Fournier (Nice),
Ch. Froeschl´e (CNRS, OCA, Nice),
C. Froeschl´e (CNRS, OCA, Nice),
A. Gombani (CNR, LADSEB, Padoue, It.),
J. Grimm (INRIA, SA),
E. Hamann (Univ. Tech. Vienne, Aut.),
J.-M. Innocent (Univ. Provence, Marseille),
J.-P. Kahane (Acad. Sciences Paris et Univ. Orsay),
E. Karatsuba (Russian Academy of Sciences, Moscow, Russia),
J. Leblond (INRIA, SA),
M. Mahjoub (LAMSIN-ENIT, Tunis),
D. Matignon (ENST, Paris),
G. M´etris (OCA, Grasse),
N.-E. Najid (Univ. Hassan II, Casablanca, Ma.),
A. Neves (Univ. Paris V),
L. Niederman (Univ. Orsay),
N. Nikolski (Univ. Bordeaux),
A. Noullez (OCA, Nice),
M. Olivi (INRIA, SA),
J.R. Partington (Univ. Leeds, GB),
J.-B. Pomet (INRIA, SA),
E.B. Saff (Univ. Vanderbilt, Nashville, USA),
F. Seyfert (INRIA, SA),
N. Sibony (Univ. Orsay),
M. Smith (Univ. York, GB),
G. Turchetti (Univ. Bologne, It.),
G. Valsecchi (Univ. Rome, It.),
J.-Y. Vinet (OCA, Nice),
P. Vitse (Univ. Laval, Qu´ebec, Ca.).