Stochastic Models, Information Theory, and Lie Groups, Volume 2: Analytic Methods and Modern Applications

458 Pages·2012·3.68 MB·English

by  Gregory S. Chirikjian (auth.)

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 Technische Universität Berlin
LosAngeles,CA,USA Berlin, Germany
DouglasCochran MauroMaggioni
ArizonaStateUniversity DukeUniversity
Phoenix,AZ,USA Durham,NC,USA
HansG.Feichtinger ZuoweiShen
UniversityofVienna NationalUniversityofSingapore
Vienna,Austria Singapore
ChristopherHeil ThomasStrohmer
GeorgiaInstituteofTechnology UniversityofCalifornia
Atlanta,GA,USA Davis,CA,USA
StéphaneJaffard YangWang
UniversityofParisXII MichiganStateUniversity
Paris,France EastLansing,MI,USA
Gregory S. Chirikjian
Stochastic Models,
Information Theory,
and Lie Groups,
Volume 2
Analytic Methods and Modern Applications
Gregory S. Chirikjian
Department of Mechanical Engineering
The Johns Hopkins University
Baltimore, MD 21218-2682
USA
[email protected]
ISBN 978-0-8176-4943-2 e-ISBN 978-0-8176-4944-9
DOI 10.1007/978-0-8176-4944-9
Springer New York Dordrecht Heidelberg London
Library of Congress Control Number: 2011941792
Mathematics Subject Classification (2010): 22E60, 53Bxx, 53C65, 58A15, 58J65, 60D05, 60H10, 70G45, 82C31,
94A15, 94A17
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To my family

ANHA Series Preface
TheAppliedandNumericalHarmonicAnalysis(ANHA)bookseriesaimstoprovidethe
engineering, mathematical, and scientific communities with significant developments in
harmonic analysis, ranging from abstract harmonic analysis to basic applications. The
titleoftheseriesreflectstheimportanceofapplicationsandnumericalimplementation,
but richness and relevance of applications and implementation depend fundamentally
on the structure and depth of theoretical underpinnings. Thus, from our point of view,
the interleaving of theory and applications and their creative symbiotic evolution is
axiomatic.
Harmonic analysis is a wellspring of ideas and applicability that has flourished, de-
veloped, and deepened over time within many disciplines and by means of creative
cross-fertilization with diverse areas. The intricate and fundamental relationship be-
tween harmonic analysis and fields such as signal processing, partial differential equa-
tions (PDEs), and image processing is reflected in our state-of-the-art ANHA series.
Ourvisionofmodernharmonicanalysisincludesmathematicalareassuchaswavelet
theory, Banach algebras, classical Fourier analysis, time-frequency analysis, and fractal
geometry, as well as the diverse topics that impinge on them.
Forexample,wavelettheorycanbeconsideredanappropriatetooltodealwithsome
basic problems in digital signal processing, speech and image processing, geophysics,
pattern recognition, biomedical engineering, and turbulence. These areas implement
the latest technology from sampling methods on surfaces to fast algorithms and com-
puter vision methods. The underlying mathematics of wavelet theory depends not only
on classical Fourier analysis, but also on ideas from abstract harmonic analysis, includ-
ingvonNeumannalgebrasandtheaffinegroup.ThisleadstoastudyoftheHeisenberg
group and its relationship to Gabor systems, and of the metaplectic group for a mean-
ingful interaction of signal decomposition methods. The unifying influence of wavelet
theory in the aforementioned topics illustrates the justification for providing a means
for centralizing and disseminating information from the broader, but still focused, area
of harmonic analysis. This will be a key role of ANHA. We intend to publish with the
scope and interaction that such a host of issues demands.
Along with our commitment to publish mathematically significant works at the
frontiers of harmonic analysis, we have a comparably strong commitment to publish
major advances in the following applicable topics in which harmonic analysis plays a
substantial role:
vii
viii ANHA Series Preface
Antennatheory Predictiontheory
Biomedicalsignalprocessing Radar applications
Digitalsignalprocessing Sampling theory
Fastalgorithms Spectralestimation
Gabor theory andapplications Speechprocessing
Imageprocessing Time-frequencyand
Numerical partial differential equations time-scaleanalysis
Wavelettheory
The above point of view for the ANHA book series is inspired by the history of
Fourier analysis itself, whose tentacles reach into so many fields.
InthelasttwocenturiesFourieranalysishashadamajorimpactonthedevelopment
of mathematics, on the understanding of many engineering and scientific phenomena,
and on the solution of some of the most important problems in mathematics and the
sciences.Historically,Fourierseriesweredevelopedintheanalysisofsomeoftheclassical
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 analysis were defined, e.g., the concept of “function.” Since
the coefficients of Fourier series are integrals, it is no surprise that Riemann integrals
were conceived to deal with uniqueness properties of trigonometric series. Cantor’s set
theory was also developed because of such uniqueness questions.
AbasicprobleminFourieranalysisistoshowhowcomplicatedphenomena,suchas
soundwaves,canbedescribedintermsofelementaryharmonics.Therearetwoaspects
of this problem: first, to find, or even define properly, the harmonics or spectrum of a
givenphenomenon,e.g.,thespectroscopyprobleminoptics;second,todeterminewhich
phenomena can be constructed from given classes of harmonics, as done, for example,
by the mechanical synthesizers in tidal analysis.
Fourier analysis is also the natural setting for many other problems in engineering,
mathematics, and the sciences. For example, Wiener’s Tauberian theorem in Fourier
analysisnotonlycharacterizesthebehavioroftheprimenumbers,butalsoprovidesthe
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 naturally with Hardy spaces in the theory
of complex variables.
Nowadays,someofthetheoryofPDEshasgivenwaytothestudyofFourierintegral
operators.Problemsinantennatheoryarestudiedintermsofunimodulartrigonometric
polynomials.ApplicationsofFourieranalysisaboundinsignalprocessing,whetherwith
the fast Fourier transform (FFT), or filter design, or the adaptive modeling inherent
in time-frequency-scale methods such as wavelet theory. The coherent states of mathe-
matical physics are translated and modulated Fourier transforms, and these are used,
in conjunction with the uncertainty principle, for dealing with signal reconstruction in
communications theory. We are back to the raison d’ˆetre of the ANHA series!
John J. Benedetto
Series Editor
University of Maryland
College Park
Preface
Preface to Volume 1
Asanundergraduatestudentatagoodengineeringschool,Ihadneverheardofstochas-
ticprocessesorLiegroups(eventhoughIdoublemajoredinMathematics).Asafaculty
member in engineering I encountered many problems where the recurring themes were
“noise” and “geometry.” When I went to read up on both topics I found fairly little at
this intersection. Now, to be certain, there are many wonderful texts on one of these
subjects or the other. And to be fair, there are several advanced treatments on their
intersection. However, for the engineer or scientists who has the modest goal of mod-
eling a stochastic (i.e., time-evolving and random) mechanical system with equations
with an eye towards numerically simulating the system’s behavior rather than proving
theorems, very few books are out there. This is because mechanical systems (such as
robots, biological macromolecules, spinning tops, satellites, automobiles, etc.) move in
multiple spatial dimensions, and the configuration space that describes allowable mo-
tions of objects made up of rigid components does not fit into the usual framework of
linear systems theory. Rather, the configuration space manifold is usually either a Lie
group or a homogeneous space.1
My mission then became clear: write a book on stochastic modeling of (possibly
complicated) mechanical systems that a well-motivated first-year graduate student or
undergraduate at the senior level in engineering or the physical sciences could pick
up and read cover-to-cover without having to carry around twenty other books. The
key point that I tried to keep in mind when writing this book was that the art of
mathematicalmodelingisverydifferentthantheartofprovingtheorems.Theemphasis
here is on “how to calculate” quantities (mostly analytically by hand and occasionally
numerically by computer) rather than “how to prove.” Therefore, some topics that are
treated at great detail in mathematics books are covered at a superficial level here, and
some concrete analytical calculations that are glossed over in mathematics books are
explained in detail here. In other words the goal here is not to expand the frontiers of
mathematics, but rather to translate known results to a broader audience.
The following quotes from Felix Klein2 in regard to the modern mathematics of his
day came to mind often during the writing process:
1The reader is not expected to know what these concepts mean at this point.
2F. Klein, Development of Mathematics in the 19th Century, translated by M. Ackerman
as part of Lie Groups: History, Frontiers and Applications, Vol. IX, Math Sci Press, 1979.
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