The Evolution of Applied Harmonic Analysis: Models of the Real World

366 Pages·2004·6.512 MB·English

by  Elena Prestini (auth.)

Applied and Numerical Harmonic Analysis
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UniversityofMaryland
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Applied and Numerical Harmonic Analysis
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E.Prestini:TheEvolutionofAppliedHarmonicAnalysis(ISBN0-8176-4125-4)
(ContinuedaftertheIndex)
Elena Pres tini
The Evolution
of
Applied Harmonic Analysis
Models ofthe Real World
Springer Science+Business Media, LLC
Elena Prestini
Department of Mathematics
University of Rome "Tor Vergata"
00133 Rome
Italy
Library of Congress Cataloging·in·Publication Data
Prestini, Elena, 1949-
The evolution of applied harmonic analysis 1 Elena Prestini
p. cm. -(Applied and numerica! harmonic analysis)
Rev. ed. of: Applicazioni dell'analisi armonica. Milan: U1rico Hoepli, 1996.
Includes bibliographical references and index.
ISBN 978-0-8176-4125-2 ISBN 978-0-8176-8140-1 (eBook)
DOI 10.1007/978-0-8176-8140-1
1. Harmonic analysis. I. Prestini, Elena, 1949-Applicazioni dell'analisi armonica. II.
I. Title
QA403.P74 2003
515' .2433-dc21 2003061722
CIP
AMS Subject Classifications: 01A45, 01A50, 42-03, 42Axx, 42Al6, 42A24, 42A38, 42Bxx, 42B05,
42Bl0
ISBN 978-0-8176-4125-2
© 2004 Springer Science+Business Media New York
Originally published by Birkhăuser Boston in 2004
Based on the Italian Edition, Applicazioni dell' analisi annonica, Ulrico Hoepli
Editare, Milan, 19 96
On the cover (tap to bottom): Wernhervon Braun, Rosalind Franklin, Martin Ryle, Jocelyn Bell,
and Joseph Fourier
Ali rights reserved. This work may not be translated or copied in whole or in part without the written
permission of the publisher (Birkhăuser Boston, c/o Springer-Verlag New York, Inc., 175 Fifth Avenue,
New York, NY 10010, USA) except forbrief excerpts in connection withreviews or scholarly analysis.
Use in connection with any form of information storage and retrieval, electronic adaptati an, computer
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The use in this publication of trade names, trademarks, service marks and similar terms, even if they
are not identified as such, is not tobe taken as an expression of opinion as to whether or not they are
subject to property rights.
(MV)
9 8 7 6 5 4 3 2 1 SPIN 10716491
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Contents
Foreword ix
Preface xiii
1 JosephFourier:TheManandtheMathematician 1
1.1 Anadventurouslife . . . . . . . . . . . . . . . . . . . . . 1
1.2 Thebeginnings . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Therevolutionary . . . . . . . . . . . . . . . . . . . . . . 4
1.4 AttheÉcoleNormale . . . . . . . . . . . . . . . . . . . . 7
1.5 FromimprisonmenttotheÉcolePolytechnique . . . . . . 9
1.6 Fourier,Napoleon,andtheEgyptiancampaign. . . . . . . 12
1.7 TheprefectofIsèreandtheAnalyticalTheoryofHeat . . . 16
1.8 Fourierandthe“HundredDays” . . . . . . . . . . . . . . 22
1.9 ReturntoParis . . . . . . . . . . . . . . . . . . . . . . . 25
2 IntroductiontoHarmonicAnalysis 31
2.1 Howtrigonometricseriescameabout:thevibratingstring . 31
2.2 Heatdiffusion . . . . . . . . . . . . . . . . . . . . . . . . 39
2.3 Fouriercoefficientsandseries . . . . . . . . . . . . . . . 43
2.4 Dirichletfunctionandtheorem . . . . . . . . . . . . . . . 46
2.5 LordKelvin,Michelson,andGibbsphenomenon . . . . . 49
vi Contents
2.6 Theconstructionofacellar . . . . . . . . . . . . . . . . . 53
2.7 TheFouriertransform . . . . . . . . . . . . . . . . . . . . 56
2.8 Dilationsandtheuncertaintyprinciple . . . . . . . . . . . 59
2.9 Translationsandinterferencefringes . . . . . . . . . . . . 59
2.10 Waves,aunifyingconceptinscience . . . . . . . . . . . . 62
3 TelecommunicationsandSpaceExploration 65
3.1 Thegreatelectricalrevolution . . . . . . . . . . . . . . . 65
3.2 Moreandmorehertz . . . . . . . . . . . . . . . . . . . . 71
3.3 TablesofFouriertransforms . . . . . . . . . . . . . . . . 76
3.4 TheDiracdeltafunctionandrelatedtopics . . . . . . . . . 79
3.5 Convolution . . . . . . . . . . . . . . . . . . . . . . . . . 81
3.6 Filters,noise,andfalsealarms . . . . . . . . . . . . . . . 84
3.7 Samplingandquantization . . . . . . . . . . . . . . . . . 87
3.8 ThediscreteFouriertransform . . . . . . . . . . . . . . . 91
3.9 ThefastFouriertransform . . . . . . . . . . . . . . . . . 93
3.10 Gaussandtheasteroids:historyoftheFFT . . . . . . . . 97
3.11 Unsteadycombustionandspacepropulsion . . . . . . . . 101
3.12 TheMarsprojectfromvonBrauntoKorolevandRubbia . 108
4 Sound,Music,andComputers 113
4.1 Understandingsound:somehistory. . . . . . . . . . . . . 113
4.2 Bats,whales,andsealions:ultrasonicreflectionsinairand
water . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
4.3 Bels,decibels,andharmonicanalysisofasingletone . . . 117
4.4 Soundprocessingbytheear:pitch,loudness,andtimbre . 119
4.5 Thetemperedscale . . . . . . . . . . . . . . . . . . . . . 122
4.6 Analysisofcomplextones:cello,clarinet,trumpet,andbass
drum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
4.7 ThreetenorsinVerdi’sRigoletto:dynamicspectra . . . . 125
4.8 Thebeginningsoftheelectronicmusicsynthesizers . . . . 127
4.9 Additivesynthesis . . . . . . . . . . . . . . . . . . . . . . 132
4.10 Subtractivesynthesisandfrequencymodulation . . . . . . 136
4.11 Besselfunctionsanddrums:thevibratingmembrane . . . 138
4.12 Convolutionandsomespecialeffects . . . . . . . . . . . . 141
4.13 Computermusicandartmusic . . . . . . . . . . . . . . . 142
5 FourierOpticsandtheSynchrotronLight 143
5.1 Thelongsearchonthenatureoflight . . . . . . . . . . . 143
5.2 Spectroscopy:acloselookatnewworlds . . . . . . . . . 147
Contents vii
5.3 Themathematicalmodelfordiffraction:theFourier
transformintwodimensions . . . . . . . . . . . . . . . . 149
5.4 Opticaltransforms . . . . . . . . . . . . . . . . . . . . . 152
5.5 X-raysandtheunfoldingoftheelectromagneticspectrum . 157
5.6 Synchrotronradiation,firstseeninthestars . . . . . . . . 160
5.7 Uniquefeaturesofthesynchrotronlight . . . . . . . . . . 163
5.8 Storagerings,amongthebiggestmachinesintheworld . . 164
5.9 Spectrumofthesynchrotronlight . . . . . . . . . . . . . 168
5.10 Brightnessandundulators. . . . . . . . . . . . . . . . . . 173
5.11 Applicationsandthefuturisticfreeelectronlaser . . . . . 174
6 X-rayCrystallography:ProteinStructureandDNA 179
6.1 CrystallographyfromStenotovonLaue . . . . . . . . . . 179
6.2 Bragg’slawandthebasicroleofsymmetries . . . . . . . 185
6.3 Howtoseetheinvisible . . . . . . . . . . . . . . . . . . . 192
6.4 Thephaseproblemandthedirectmethods . . . . . . . . . 196
6.5 Briefhistoryofgenetics. . . . . . . . . . . . . . . . . . . 197
6.6 DNAthroughx-rays . . . . . . . . . . . . . . . . . . . . 202
6.7 Proteinstructure:workforacentury . . . . . . . . . . . . 206
6.8 Seeingthehydrogenatom . . . . . . . . . . . . . . . . . 210
6.9 Visionatthenanosecond:atime-resolvedstructure . . . . 211
7 TheRadonTransformandComputerizedTomography 215
7.1 Computerizedaxialtomography,atechniquefrom
the1970s . . . . . . . . . . . . . . . . . . . . . . . . . . 215
7.2 Beyondclassicalradiology . . . . . . . . . . . . . . . . . 217
7.3 CTversusMRI . . . . . . . . . . . . . . . . . . . . . . . 222
7.4 Projections . . . . . . . . . . . . . . . . . . . . . . . . . 223
7.5 TheRadontransform . . . . . . . . . . . . . . . . . . . . 224
7.6 Inverting the Radon transform by convolution and back
projections. . . . . . . . . . . . . . . . . . . . . . . . . . 227
7.7 CTimages . . . . . . . . . . . . . . . . . . . . . . . . . . 231
7.8 Localtomographyandmicrotomography . . . . . . . . . 231
8 NuclearMagneticResonance:ImagingandSpectroscopy 237
8.1 “Weareallradiostations” . . . . . . . . . . . . . . . . . 237
8.2 Nuclearspinandmagneticmoment . . . . . . . . . . . . 245
8.3 Theswingandthephenomenonofresonance . . . . . . . 247
8.4 Drivingaresonance:theLarmorfrequency . . . . . . . . 248
8.5 Relaxationprocess . . . . . . . . . . . . . . . . . . . . . 250
viii Contents
8.6 Relaxationtimes . . . . . . . . . . . . . . . . . . . . . . 251
8.7 NMRspectroscopy:chemicalshiftandcouplingconstant . 255
8.8 Prionprotein . . . . . . . . . . . . . . . . . . . . . . . . 259
8.9 CollectingNMRspectra:continuouswaveversuspulses . 261
8.10 TheroadtoNMRimagingintheworkofDamadian . . . . 263
8.11 Theprincipleofimageformation . . . . . . . . . . . . . . 266
8.12 MRI-guidedinterventions . . . . . . . . . . . . . . . . . . 270
8.13 FunctionalMRI . . . . . . . . . . . . . . . . . . . . . . . 272
9 RadioastronomyandModernCosmology 275
9.1 Astronomicalobservations:
“sixthousandyearsforawitness” . . . . . . . . . . . . . 275
9.2 Howradioastronomywasborn . . . . . . . . . . . . . . . 281
9.3 Themost“mortifying"episodeofEnglishnavalhistory . . 284
9.4 Radiowavesandradiotelescopes . . . . . . . . . . . . . . 286
9.5 Resolvingpower . . . . . . . . . . . . . . . . . . . . . . 287
9.6 Interferometry . . . . . . . . . . . . . . . . . . . . . . . . 290
9.7 Theprincipleofimageformationinaradiotelescope . . . 292
9.8 Aperturesynthesis . . . . . . . . . . . . . . . . . . . . . . 295
9.9 SynthesisbyEarth’srotation . . . . . . . . . . . . . . . . 298
9.10 Quasars,pulsars,andtheBigBang . . . . . . . . . . . . . 300
9.11 A“flat”forever-expandingUniverse . . . . . . . . . . . . 306
9.12 Searchforextraterrestrialintelligence . . . . . . . . . . . 308
Appendix 311
Bibliography 315
Index 337
Foreword
Two hundred years ago Baron Jean Baptiste Joseph Fourier (1768–1830)
championedamathematicalideathatwastohaveaprofoundandenduring
influencefarbeyondwhatcouldbeimaginedatthetime;understandingthe
significanceofhisworkrequiressomehistoricalperspective.
Arithmetic and trigonometry were already well polished. For example,
the trigonometrical table published by Claudius Ptolemy (2nd c. A.D.) in
the days of the emperors Hadrian and Antoninus was more precise and
more finely tabulated than the four-figure table of sines used today in
schools. Euclidean geometry was thoroughly familiar. Calculus, starting
with Archimedes (287–212 B.C.), who had determined the volume of a
spherebythelimitingprocessnowfamiliarasintegration,hadbeengiven
its modern formulation by Gottfried Wilhelm Leibniz (1646–1716) and
Isaac Newton (1642–1727). Calculus became a mighty tool of astronomy
inexplainingthemovementoftheEarthandplanetsandhadanimpacton
philosophybyitssuccessinshowingthatnumerouspuzzlingfeaturesofthe
Universe were explicable by deduction from a few physical premises: the
threelawsofmotionandthelawofgravitation.Butintegralanddifferential
calculushadnotbeenbroughttoperfection;theysufferedfromlimitations
whenconfrontedwithentitiesthatweredeemednonintegrableornondiffer-
entiableandfromdifficultiesinhandlingcertainnotionsinvolvinginfinity
andinfinitesimals.