Table Of ContentApplied and Numerical Harmonic Analysis
Christopher Heil
Metrics, Norms,
Inner Products,
and Operator
Theory
Applied and Numerical Harmonic Analysis
Series Editor
John J. Benedetto
University of Maryland
College Park, MD, USA
Editorial Advisory Board
Akram Aldroubi Gitta Kutyniok
Vanderbilt University Technische Universität Berlin
Nashville, TN, USA Berlin, Germany
Douglas Cochran Mauro Maggioni
Arizona State University Duke University
Phoenix, AZ, USA Durham, NC, USA
Hans G. Feichtinger Zuowei Shen
University of Vienna National University of Singapore
Vienna, Austria Singapore, Singapore
Christopher Heil Thomas Strohmer
Georgia Institute of Technology University of California
Atlanta, GA, USA Davis, CA, USA
Stéphane Jaffard Yang Wang
University of Paris XII Michigan State University
Paris, France East Lansing, MI, USA
Jelena Kovačević
Carnegie Mellon University
Pittsburgh, PA, USA
More information about this series at http://www.springer.com/series/4968
Christopher Heil
Metrics, Norms, Inner
Products, and Operator
Theory
Christopher Heil
Schoolof Mathematics
Georgia Institute of Technology
Atlanta, GA,USA
ISSN 2296-5009 ISSN 2296-5017 (electronic)
AppliedandNumerical Harmonic Analysis
ISBN978-3-319-65321-1 ISBN978-3-319-65322-8 (eBook)
https://doi.org/10.1007/978-3-319-65322-8
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In memory, Christopher Andrew Heil, 1920–2011
ANHA Series Preface
TheAppliedandNumericalHarmonicAnalysis(ANHA)bookseriesaimstoprovide
theengineering,mathematical,andscientificcommunitieswithsignificantdevelop-
mentsinharmonicanalysis,rangingfromabstractharmonicanalysistobasicappli-
cations.Thetitleoftheseriesreflectstheimportanceofapplicationsandnumerical
implementation,butrichnessandrelevanceofapplicationsandimplementationde-
pendfundamentallyonthestructureanddepthoftheoreticalunderpinnings.Thus,
fromourpointofview,theinterleavingoftheoryandapplicationsandtheircreative
symbioticevolutionisaxiomatic.
Harmonicanalysisisawellspringofideasandapplicabilitythathasflourished,
developed,and deepenedovertime within manydisciplines and by means of cre-
ativecross-fertilizationwith diverseareas.The intricateand fundamentalrelation-
shipbetweenharmonicanalysisandfieldssuchassignalprocessing,partialdiffer-
ential equations (PDEs), and image processing is reflected in our state-of-the-art
ANHAseries.
Our vision of modern harmonic analysis includes mathematical areas such as
wavelettheory,Banachalgebras,classicalFourieranalysis,time-frequencyanalysis,
andfractalgeometry,aswellasthediversetopicsthatimpingeonthem.
Forexample,wavelettheorycanbeconsideredanappropriatetooltodealwith
somebasicproblemsindigitalsignalprocessing,speechandimageprocessing,geo-
physics, pattern recognition, biomedical engineering, and turbulence. These areas
implement the latest technology from sampling methods on surfaces to fast algo-
rithmsandcomputervisionmethods.Theunderlyingmathematicsofwavelettheory
dependsnotonlyonclassicalFourieranalysis,butalsoonideasfromabstracthar-
monic analysis, including von Neumannalgebras and the affine group.This leads
toastudyoftheHeisenberggroupanditsrelationshiptoGaborsystems,andofthe
metaplectic group for a meaningful interaction of signal decomposition methods.
Theunifyinginfluenceofwavelettheoryintheaforementionedtopicsillustratesthe
justification for providinga means for centralizing and disseminating information
fromthebroader,butstillfocused,areaofharmonicanalysis.Thiswillbeakeyrole
of ANHA. We intend to publish with the scope and interaction that such a host of
issuesdemands.
Alongwithourcommitmentto publishmathematicallysignificantworksatthe
frontiersofharmonicanalysis,wehaveacomparablystrongcommitmenttopublish
vii
viii ANHASeriesPreface
majoradvancesinthefollowingapplicabletopicsinwhichharmonicanalysisplays
asubstantialrole:
Antennatheory Predictiontheory
Biomedicalsignalprocessing Radarapplications
Digitalsignalprocessing Samplingtheory
Fastalgorithms Spectralestimation
Gabortheoryandapplications Speechprocessing
Imageprocessing Time-frequencyand
Numericalpartialdifferentialequations time-scaleanalysis
Wavelettheory
TheabovepointofviewfortheANHAbookseriesisinspiredbythehistoryof
Fourieranalysisitself,whosetentaclesreachintosomanyfields.
In the last two centuriesFourier analysis has had a major impact on the devel-
opment of mathematics, on the understanding of many engineering and scientific
phenomena,andonthesolutionofsomeofthemostimportantproblemsinmathe-
maticsandthesciences.Historically,Fourierseriesweredevelopedintheanalysis
of some of the classical PDEs of mathematical physics; these series were used to
solve such equations. In order to understandFourier series and the kinds of solu-
tionstheycouldrepresent,someofthemostbasicnotionsofanalysisweredefined,
e.g.,theconceptof“function.”SincethecoefficientsofFourierseriesareintegrals,
it is no surprise that Riemann integrals were conceived to deal with uniqueness
propertiesoftrigonometricseries. Cantor’sset theorywasalso developedbecause
ofsuchuniquenessquestions.
A basic problem in Fourier analysis is to show how complicated phenomena,
suchassoundwaves,canbedescribedintermsofelementaryharmonics.Thereare
twoaspectsofthisproblem:first,tofind,orevendefineproperly,theharmonicsor
spectrumofagivenphenomenon,e.g.,thespectroscopyprobleminoptics;second,
todeterminewhichphenomenacanbeconstructedfromgivenclassesofharmonics,
asdone,forexample,bythemechanicalsynthesizersintidalanalysis.
Fourieranalysisisalsothenaturalsettingformanyotherproblemsinengineer-
ing, mathematics, and the sciences. For example, Wiener’s Tauberian theorem in
Fourieranalysisnotonlycharacterizesthebehavioroftheprimenumbers,butalso
provides the proper notion of spectrum for phenomena such as white light; this
latterprocessleadsto theFourieranalysisassociatedwith correlationfunctionsin
filtering and predictionproblems, and these problems, in turn, deal naturally with
Hardyspacesinthetheoryofcomplexvariables.
Nowadays, some of the theory of PDEs has given way to the study of Fourier
integral operators. Problems in antenna theory are studied in terms of unimodu-
lar trigonometric polynomials. Applications of Fourier analysis abound in signal
processing, whether with the fast Fourier transform (FFT), or filter design, or the
adaptivemodelinginherentintime-frequency-scalemethodssuchaswavelettheory.
The coherentstates of mathematicalphysicsare translated and modulatedFourier
ANHASeriesPreface ix
transforms, and these are used, in conjunction with the uncertainty principle, for
dealing with signal reconstruction in communications theory. We are back to the
raisond’eˆtreoftheANHAseries!
UniversityofMaryland JohnJ.Benedetto
CollegePark SeriesEditor
Contents
Preface ....................................................... xv
1 Notation and Preliminaries ............................... 1
1.1 Numbers .............................................. 1
1.2 Sets .................................................. 2
1.3 Functions.............................................. 4
1.4 Cardinality ............................................ 5
1.5 Sequences ............................................. 10
1.6 Suprema and Infima .................................... 12
1.7 Convergent Sequences of Numbers ........................ 14
1.8 Infinite Series of Numbers ............................... 16
1.9 Differentiation and the Riemann Integral .................. 18
1.10 Vector Spaces.......................................... 20
1.11 Span and Independence ................................. 26
2 Metric Spaces ............................................ 31
2.1 Metrics ............................................... 31
2.2 Convergence and Completeness........................... 42
2.3 Topology in Metric Spaces............................... 53
2.4 Closed Sets ............................................ 59
2.5 Accumulation Points and Boundary Points ................ 62
2.6 Closure, Density, and Separability ........................ 66
2.7 The Cantor Set ........................................ 73
2.8 Compact Sets in Metric Spaces........................... 75
2.9 Continuity for Functions on Metric Spaces................. 83
2.10 Urysohn’s Lemma ...................................... 90
2.11 The Baire Category Theorem ............................ 94
3 Norms and Banach Spaces................................ 99
3.1 The Definition of a Norm................................ 99
3.2 The ℓp Spaces.......................................... 103
xi
