Distributions in the physical and engineering sciences. 2, Linear and nonlinear dynamics in continuous media / Alexander I. Saichev; Wojbor A. Woyczynski

427 Pages·2013·3.44 MB·English

by  Saičev

Applied and Numerical Harmonic Analysis
Alexander I. Saichev
Wojbor A. Woyczyn´ski
Distributions
in the Physical
and Engineering
Sciences, Volume 2
Linear and Nonlinear Dynamics in
Continuous Media

Applied and Numerical Harmonic Analysis
SeriesEditor
JohnJ.Benedetto
UniversityofMaryland
CollegePark,MD,USA
EditorialAdvisoryBoard
AkramAldroubi JelenaKovacˇevic´
VanderbiltUniversity CarnegieMellonUniversity
Nashville,TN,USA Pittsburgh,PA,USA
AndreaBertozzi GittaKutyniok
UniversityofCalifornia TechnischeUniversita¨tBerlin
LosAngeles,CA,USA Berlin,Germany
DouglasCochran MauroMaggioni
ArizonaStateUniversity DukeUniversity
Phoenix,AZ,USA Durham,NC,USA
HansG.Feichtinger ZuoweiShen
UniversityofVienna NationalUniversityofSingapore
Vienna,Austria Singapore,Singapore
ChristopherHeil ThomasStrohmer
GeorgiaInstituteofTechnology UniversityofCalifornia
Atlanta,GA,USA Davis,CA,USA
Ste´phaneJaffard YangWang
UniversityofParisXII MichiganStateUniversity
Paris,France EastLansing,MI,USA
Forfurthervolumes:
http://www.springer.com/series/4968
Alexander I. Saichev • Wojbor A. Woyczyn´ski
Distributions in the Physical
and Engineering Sciences,
Volume 2
Linear and Nonlinear Dynamics
in Continuous Media
AlexanderI.Saichev WojborA.Woyczyn´ski
DepartmentofManagement, DepartmentofMathematics,
Technology,andEconomics AppliedMathematicsandStatistics,andCenter
ETHZu¨rich forStochasticandChaoticProcessesinScience
Zu¨rich,Switzerland andTechnology
CaseWesternReserveUniversity
DepartmentofRadioPhysics
Cleveland,OH,USA
UniversityofNizhniyNovgorod
NizhniyNovgorod,Russia
Additionalmaterialtothisbookcanbedownloadedfromhttp://extras.springer.com
ISBN978-0-8176-3942-6 ISBN978-0-8176-4652-3(eBook)
DOI10.1007/978-0-8176-4652-3
SpringerNewYorkHeidelbergDordrechtLondon
LibraryofCongressControlNumber:96039028
MathematicsSubjectClassification(2010):31-02,31Axx,31Bxx,35-02,35Dxx,35Jxx,35Kxx,35Lxx,35Qxx,
70-02,76-02,76Lxx,76Nxx,76Sxx
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ANHA Series Preface
TheAppliedandNumericalHarmonicAnalysis(ANHA)bookseriesaimstopro-
vide the engineering, mathematical, and scientific communities with significant
developments in harmonic analysis, ranging from abstract harmonic analysis to
basic applications. The title of the series reflects the importance of applications
and numerical implementation, but richness and relevance of applications and
implementation depend fundamentally on the structure and depth of theoretical
underpinnings.Thus,fromourpointofview,theinterleavingoftheoryandappli-
cationsandtheircreativesymbioticevolutionisaxiomatic.
Harmonic analysis is a wellspring of ideas and applicability that has flour-
ished, developed, and deepened over time within many disciplines and by means
ofcreativecross-fertilizationwithdiverseareas.Theintricateandfundamentalre-
lationshipbetweenharmonicanalysisandfieldssuchassignalprocessing,partial
differential equations (PDEs), and image processing is reflected in our state-of-
theartANHAseries.
Our visionof modernharmonic analysis includesmathematicalareas suchas
wavelettheory,Banachalgebras,classicalFourieranalysis,time–frequencyanal-
ysis,andfractalgeometry,aswellasthediversetopicsthatimpingeonthem.
For example, wavelet theory can be considered an appropriate tool to deal
with some basic problems in digital signal processing, speech and image pro-
cessing,geophysics,patternrecognition,biomedicalengineering,andturbulence.
These areas implement the latest technology from sampling methods on surfaces
to fast algorithms and computer vision methods. The underlying mathematics of
wavelet theory depends not only on classical Fourier analysis, but also on ideas
from abstract harmonic analysis, including von Neumann algebras and the affine
group.ThisleadstoastudyoftheHeisenberggroupanditsrelationshiptoGabor
systems, and of the metaplectic group for a meaningful interaction of signal de-
composition methods. The unifying influence of wavelet theory in the aforemen-
tionedtopicsillustratesthejustificationforprovidingameansforcentralizingand
disseminating information from the broader, but still focused, area of harmonic
analysis. This will be a key role of ANHA. We intend to publish the scope and
interactionthatsuchahostofissuesdemands.
v
vi ANHA Series Preface
Alongwithourcommitmenttopublishmathematicallysignificantworksatthe
frontiersofharmonicanalysis,wehaveacomparablystrongcommitmenttopub-
lishmajoradvancesinthefollowingapplicabletopicsinwhichharmonicanalysis
playsasubstantialrole:
Biomedicalsignalprocessing Numericalpartialdifferentialequations
Compressivesensing Predictiontheory
Communicationsapplications Radarapplications
Datamining/machinelearning Samplingtheory
Digitalsignalprocessing Spectralestimation
Fastalgorithms Speechprocessing
Gabortheoryandapplications Time–frequencyandtime-scaleanalysis
Imageprocessing Wavelettheory
The above point of view for the ANHA book series is inspired by the history
ofFourieranalysisitself,whosetentaclesreachintosomanyfields.
In the last two centuries, Fourier analysis has had a major impact on the de-
velopment of mathematics, on the understanding of many engineering and sci-
entific phenomena, and on the solution of some of the most important problems
in mathematics and the sciences. Historically, Fourier series were developed in
the analysis of some of the classical PDEs of mathematical physics; these series
were used to solve such equations. In order to understand Fourier series and the
kinds of solutions they could represent, some of the most basic notions of analy-
sisweredefined,e.g.,theconceptof“function”.SincethecoefficientsofFourier
seriesareintegrals,itisnosurprisethatRiemannintegralswereconceivedtodeal
with uniqueness properties of trigonometric series. Cantor’s set theory was also
developedbecauseofsuchuniquenessquestions.
A basic problem in Fourier analysis is to show how complicated phenomena,
such as sound waves, can be described in terms of elementary harmonics. There
aretwoaspectsofthisproblem:first,tofind,orevendefineproperly,theharmon-
icsorspectrumofagivenphenomenon,e.g.,thespectroscopyprobleminoptics;
second,todeterminewhichphenomenacanbeconstructedfromgivenclassesof
harmonics,asdone,e.g.,bythemechanicalsynthesizersintidalanalysis.
Fourier analysis is also the natural setting for many other problems in engi-
neering, mathematics, and the sciences. For example, Wiener’s Tauberian theo-
reminFourieranalysisnotonlycharacterizesthebehavioroftheprimenumbers,
but also provides the proper notion of spectrum for phenomena such as white
light; this latter process leads to the Fourier analysis associated with correlation
functions in filtering and prediction problems, and these problems, in turn, deal
naturallywithHardyspacesinthetheoryofcomplexvariables.
ANHA Series Preface vii
Nowadays,someofthetheoryofPDEshasgivenwaytothestudyofFourier
integral operators. Problems in antenna theory are studied in terms of unimodu-
lar trigonometric polynomials. Applications of Fourier analysis abound in signal
processing,whetherwiththefastFouriertransform(FFT),orfilterdesign,orthe
adaptivemodelinginherentintime–frequency-scalemethodssuchaswaveletthe-
ory. The coherent states of mathematical physics are translated and modulated
Fouriertransforms,andtheseareused,inconjunctionwiththeuncertaintyprinci-
ple,fordealingwithsignalreconstructionincommunicationstheory.Weareback
totheraisond’eˆtreoftheANHAseries!
UniversityofMaryland JohnJ.Benedetto
CollegePark

To Tanya and Liz—
with love and respect