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Lokenath Debnath Firdous Ahmad Shah Wavelet Transforms and Their Applications Second Edition Lokenath Debnath • Firdous Ahmad Shah Wavelet Transforms and Their Applications Second Edition Z Z 1 1 (cid:2) (cid:3) (cid:2) (cid:3) dbda W f .a;b/ W g .a;b/ DC hf;gi § § a2 § (cid:2)1 (cid:2)1 LokenathDebnath FirdousAhmadShah DepartmentofMathematics DepartmentofMathematics UniversityofTexas, UniversityofKashmir PanAmerican Anantnag,JammuandKashmir,India Edinburg,TX,USA ISBN978-0-8176-8417-4 ISBN978-0-8176-8418-1(eBook) DOI10.1007/978-0-8176-8418-1 SpringerNewYorkHeidelbergDordrechtLondon LibraryofCongressControlNumber:2014953313 MathematicsSubjectClassification(2010):01A,35Q30,35Q53,35K05,42A16,42A38,42A55,42A65, 42C05,42C10 ©SpringerScience+BusinessMediaNewYork2002,2015 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped.Exemptedfromthislegalreservationarebriefexcerptsinconnection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher’slocation,initscurrentversion,andpermissionforusemustalwaysbeobtainedfromSpringer. PermissionsforusemaybeobtainedthroughRightsLinkattheCopyrightClearanceCenter.Violations areliabletoprosecutionundertherespectiveCopyrightLaw. Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant protectivelawsandregulationsandthereforefreeforgeneraluse. While the advice and information in this book are believed to be true and accurate at the date of publication,neithertheauthorsnortheeditorsnorthepublishercanacceptanylegalresponsibilityfor anyerrorsoromissionsthatmaybemade.Thepublishermakesnowarranty,expressorimplied,with respecttothematerialcontainedherein. Printedonacid-freepaper SpringerispartofSpringerScience+BusinessMedia(www.birkhauser-science.com) Preface to the Second Edition Ateachercannevertrulyteachunlessheisstilllearninghimself.Alampcanneverlight anotherlampunlessitcontinuestoburnitsownflame.Theteacherwhohascometotheend ofthissubject,whohasnolivingtrafficwithhisknowledgebutmerelyrepeatshislessons tohisstudents,canonlyloadtheirminds;hecannotquickenthem. RabindranathTagore NobelPrizeWinnerforLiterature(1913) Thefirsteditionofthisbookwasfirstpublishedin2002underthesoleauthorship ofDr.LokenathDebnath.Itwaswellreceivedandusedasa seniorundergraduate and first year graduate level text and reference in the USA and abroad for the last12years.Wereceivedvariouscommentsandsuggestionsfrommanystudents, faculty,andresearchersaroundtheworld.Thesecommentsandcriticismshavebeen veryhelpful,beneficial,andencouraging.Thesecondeditionistheresultofthese suggestionsandcomments. The selection, arrangement, and presentation of the material in this edition have carefully been made based on our past and present teaching, research, and professionalexperience.In particular,this bookhasevolvedfromregularlyteach- ing courses in wavelet transforms, signal analysis, differential equations, applied mathematics, and advancedengineeringmathematics overmany years to students ofmathematicsandengineeringintheUSAandabroad.Itisessentiallydesignedto cover advanced mathematical methods for science and engineering students with heavy emphasis to many different and varied applications. It differs from many textbookswithsimilartitlesduetomajoremphasisplacedonnumeroustopicsand systematic developmentof the underlying theory before making applications and inclusionofmanynewandmoderntopicssuchasmultiresolutionanalysisandcon- structionofwavelets,extensionsofmultiresolutionanalysis,Gabortransformsand time–frequencysignalanalysis,theWigner–Villedistribution,andtime–frequency signal analysis. An attempt has also been made to provide a modern approach to fractals, turbulence, and Newland’s harmonic wavelets. Some of these new topics includedin this secondeditionare notfoundin other textsand researchreference books. v vi PrefacetotheSecondEdition This extensively revised second edition preserves the basic content, style, and formatofthefirsteditionpublishedin2002.Aswiththepreviousedition,thisbook has been revised primarilyas a comprehensivetext for senior undergraduatesand beginning graduate students and a research reference for professionals in mathematics, science, engineering, and other applied sciences. With basic prerequisites of calculus and ordinary differential equations, the main goal of the edition is to develop required analytical knowledge and skills on the part of the reader, rather thanfocuson the importanceof moreabstractformulationwith fullmathematical rigor.Indeed,ourmajoremphasisistoprovideanaccessibleworkingknowledgeof theanalyticalandcomputationalmethodswithproofsrequiredinpureandapplied mathematics,physics,andengineering. Mathematics, science, and engineering students need to gain a sound knowledge of mathematical and computational skills. This book provides these by the systematic development of underlying theory with varied applications and by provision of carefully selected fully worked-out examples combined with their extensions and refinements through additions of a large set of a wide variety of exercises at the end of each chapter. Numerous standard and challenging topics, applications,worked-outexamples,andexercisesareincludedinthiseditionsothat theystimulateresearchinterestamongseniorundergraduatesandgraduatestudents. Anotherspecialfeatureofthisbookistoincludesufficientmoderntopicswhichare vital prerequisitesfor subsequentadvancedcourses and research in mathematical, physical,andengineeringsciences. Readers familiar with the previous edition will notice many minor changes and numerous major ones in this edition. In general, changes have been made to modernizethe contentsand to improvethe expositionsand clarity of the previous edition to include additional materials, proofs, and comments as well as many examplesofapplicationsandexercises,andinsomecasestoentirelyrewritemany sections. There is plenty of material in the book for long course, seminars, or workshops.Someofthematerialsneednotbecoveredinacoursework,seminars, or workshops and can be left for the readers to study on their own. This edition containsa collectionoffullyworked-outexamplesandchallengingexerciseswith detailed answers and hints to many selected exercises. We have also updated the bibliographyand corrected typographicalerrors. Major changesand additions includethefollowing: 1. In Chap.1, a brief historical introduction has been completely revised and expanded to include many new topics including the fractional Fourier transform, the fractional wavelet transform, the discrete wavelet transform, and the complex wavelet transform as well as the construction of wavelet bases in various spaces other than R and several new extensions of the original multiresolutionanalysis.Inaddition,thelasttwodecadeshaveseentremendous activityinthedevelopmentofnewmathematicalandcomputationaltoolsbased on multiscale ideassuchas steerablewavelets, wedgelets, beamlets,bandlets, ridgelets, curvelets, contourlets,surfacelets, shearlets, and platelets. This new historicaldiscussionofthesubjecthasbeenincludedtohelpthereaderseethe PrefacetotheSecondEdition vii directionsinwhichthesubjecthasdevelopedandthenewmajorcontributions toitsrecentdevelopments. 2. In order to make the book self-contained, Chap.2 on Hilbert spaces and orthonormal systems with applications, and Chap.3 on the theory of Fourier transforms and their diverse applications have been presented in great detail. TwonewtopicsdealingwithdiscreteFouriertransforms(DFT)andfastFourier transforms(FFT)havebeenaddedtoChap.3. 3. The Gabor transform and the Wigner–Ville distribution with time–frequency signal analysis are the major topics of Chaps.4 and 5. Included are the Zak transform and its basic properties including the Balian–Low theorem and applicationsforstudyingtheorthogonalityandcompletenessofGaborframes in the critical case. The relationship between the Wigner–Ville distribution and ambiguity functions is investigated with radar signal analysis. Recent generalizationsoftheWigner–Villedistributionsarebrieflydescribed. 4. WavelettransformsandtheirbasicpropertiesarediscussedinChap.6insome detail. The discrete wavelet transforms and orthogonalwavelets are included inChap.6.Chapter9dealswithNewland’sharmonicwaveletsandtheirbasic properties.Specialattentionisgiventopropertiesofharmonicscalingfunctions andParseval’sformulaforharmonicwavelets. 5. InChap.7,multiresolutionanalysiswithexamplesandconstructionofwavelets are describedinsomedetail.Thischapterincludesbasicpropertiesofscaling functions, bases of orthonormal wavelets, and construction of orthonormal wavelets. Special attention is given to Daubechies’ wavelet and algorithms, discrete wavelet transforms, and Mallat’s pyramid algorithms. In order to modernize the content of the book, two new major extensions of the original multiresolutionanalysisconsistingofp-multiresolutionanalysisonthepositive half-line and nonuniform multiresolution have been included in this new Chap.8. 6. ThefinalchapterdealswithabriefdiscussionoftheclassicalFouriertransform treatmentofturbulencebasedontheNavier–Stokesequationsandtheequation of continuity. This is followed by a new treatment of certain aspects of turbulence based on fractals, multifractals, and singularities in turbulence. Includedinthischapteristhemodernapproachtoturbulenceusingthewavelet transform analyses. Special attention, in some detail, is given to Farge’s and Meneveau’s wavelet transform analyses of turbulent flows with the adaptive waveletmethodofcomputationandanalysisofturbulence. 7. Thebookoffersadetailedandclearexplanationofeveryconceptandmethod that is introduced, accompanied by carefully selected worked-out examples withspecialemphasisbeinggiventothosetopicsinwhichstudentsexperience difficulty. 8. A wide variety of modern topics has been selected from areas of wavelet analysis,signalanalysis,ordinaryandpartialdifferentialequations,turbulence, multiresolutionanalysisandconstructionofwavelets,andNewland’sharmonic wavelets. viii PrefacetotheSecondEdition 9. Thebookisorganizedwithsufficientflexibilityinteachingcoursesordirecting seminars and workshops to enable instructors to select topics and chapters appropriatetocoursesofdifferinglengths,emphases,andlevelsofdifficulty. 10. A wide spectrum of exercises has been carefully chosen and included at the end of each chapter, so the reader may further develop both analytical and computational skills in the theory and applications of wavelets, wavelet transformanalysis,andmultiresolutionanalysisandmaygainadeeperinsight intothosesubject. 11. Thebibliographyhasbeencompletelyrevisedandupdated.Manynewresearch papers and standard books have been added to the bibliography to stimulate new interest in future advanced study and research. The index has also been completelyrevisedinordertoincludeawidevarietyoftopics. 12. Thebookprovidesinformationthatputsthereaderattheforefrontofadvanced studyandcurrentresearch. With many improvements and numerous challenging topics, applications, and exercises, we hope this edition will continue to be a useful textbook for students as well as a research reference for professionals in mathematics, science, and engineering. Itisourpleasuretoexpressourgratefulthankstomanyfriends,colleagues,and studentsaroundtheworldwhoofferedtheirsuggestionsandhelpatvariousstages of the preparation of the book. In spite of the best efforts of everyone involved, sometypographicalerrorsdoubtlessremain.Finally,wewishtoexpressourthanks toMrs.DanielleWalker,AssociateEditor,andthestaffofBirkhäuser-Springerfor theirhelpandcooperation. Edinburg,TX,USA LokenathDebnath Anantnag,JammuandKashmir,India FirdousAhmadShah Preface to the First Edition Overview Historically, the concept of “ondelettes” or “wavelets” originated from the study of time–frequency signal analysis, wave propagation, and sampling theory. One of the main reasons for the discovery of wavelets and wavelet transforms is that the Fourier transform analysis does not contain the local information of signals. So the Fourier transform cannot be used for analyzing signals in a joint time and frequency domain. In 1982, Jean MorIet, in collaboration with a group of French engineers, first introduced the idea of wavelets as a family of functions constructedbyusingtranslationanddilationofasinglefunction,calledthemother wavelet, for the analysis of nonstationary signals. However, this new concept can be viewed as the synthesis of various ideas originating from different disciplines including mathematics (Calderón–Zygmund operators and Littlewood–Paley theory),physics(coherentstatesinquantummechanicsandtherenormalizationgroup), andengineering(quadraticmirrorfilters,sidebandcodinginsignalprocessing,and pyramidalalgorithmsinimageprocessing). Wavelet analysis is an exciting new method for solving difficult problems in mathematics,physics,andengineering,withmodernapplicationsasdiverseaswave propagation, data compression, image processing, pattern recognition, computer graphics,thedetectionofaircraftandsubmarines,andimprovementinCATscans andothermedicalimagetechnology.Waveletsallowcomplexinformationsuchas music,speech,images,andpatternstobedecomposedintoelementaryforms,called thefundamentalbuildingblocks,atdifferentpositionsandscalesandsubsequently reconstructedwithhighprecision.Withevergreaterdemandformathematicaltools toprovideboththeoryandapplicationsforscienceandengineering,theutilityand interestofwaveletanalysisseemmoreclearlyestablishedthanever.Keepingthese things in mind, our main goal in this modest book has been to provide both a systematicexpositionofthebasicideasandresultsofwavelettransformsandsome applicationsintime–frequencysignalanalysisandturbulence. ix