Atlantis Studies in Mathematics for Engineering and Science Series Editor: Charles K. Chui Radomir S. Stanković · Paul L. Butzer Ferenc Schipp · William R. Wade Authors/Co-Authors Dyadic Walsh Analysis from 1924 Onwards Walsh-Gibbs-Butzer Dyadic Differentiation in Science Volume 1 Foundations A Monograph Based on Articles of the Founding Authors, Reproduced in Full in Collaboration with the Co-Authors Weiyi Su, Yasushi Endow, Sándor Fridli, Boris I. Golubov, Franz Pichler Kees (C.W.) Onneweer Atlantis Studies in Mathematics for Engineering and Science Volume 12 Series editor Charles K. Chui, Department of Statistics, Stanford University, Stanford, California, USA Aims and scope of the series The series ‘Atlantis StudiesinMathematics forEngineering andScience’(AMES) publishes high quality monographs in applied mathematics, computational mathematics, and statistics that have the potential to make a significant impact on the advancement of engineering and science on the one hand, and economics and commerce on the other. We welcome submission of book proposals and manuscripts from mathematical scientists worldwide who share our vision of mathematics as the engine of progress in the disciplines mentioned above. For more information on this series and our other book series, please visit our website at: www.atlantis-press.com/publications/books Atlantis Press 8, square des Bouleaux 75019 Paris, France More information about this series at http://www.springer.com/series/10071 ć Radomir S. Stankovi Paul L. Butzer (cid:129) Ferenc Schipp William R. Wade (cid:129) Authors/Co-Authors Dyadic Walsh Analysis from 1924 Onwards Walsh-Gibbs-Butzer Dyadic Differentiation in Science Volume 1 Foundations A Monograph Based on Articles of the Founding Authors, Reproduced in Full in Collaboration with the Co-Authors Weiyi Su Yasushi Endow Sándor Fridli Boris I. Golubov Franz Pichler Kees (C.W.) Onneweer Authors Co-Authors Radomir S. Stanković Weiyi Su Faculty of Electronic Engineering Mathematics Nis NanjingUniversity Serbia Nanjing China PaulL. Butzer Lehrstuhl A für Mathematik YasushiEndow RWTH Aachen ChuoUniversity Aachen Tokyo Germany Japan Ferenc Schipp SándorFridli Numerical Analysis Numerical Analysis EötvösLorándUniversity EötvösLorándUniversity Budapest Budapest Hungary Hungary William R. Wade Boris I.Golubov Mathematics andComputer Science Chairof Higher Mathematics BIOLA University Moscow Institute of Physics LaMiranda, California andTechnology(State University) USA Dolgoproudny,Moscow Region Russia FranzPichler System Theory JohannesKepler University, Linz Linz Austria Kees (C.W.)Onneweer Department ofMathematics andStatistics University of NewMexico Albuquerque USA ISSN 1875-7642 ISSN 2467-9631 (electronic) Atlantis Studies inMathematics for Engineering andScience ISBN978-94-6239-159-8 ISBN978-94-6239-160-4 (eBook) DOI 10.2991/978-94-6239-160-4 LibraryofCongressControlNumber:2015954975 ©AtlantisPressandtheauthor(s)2015 Thisbook,oranypartsthereof,maynotbereproducedforcommercialpurposesinanyformorbyany means, electronic or mechanical, including photocopying, recording or any information storage and retrievalsystemknownortobeinvented,withoutpriorpermissionfromthePublisher. Printedonacid-freepaper To thememoryof JosephLeonard Walshand JamesEdmundGibbs Preface ThecentralthemeofthisbookistheGibbsderivative,abroadclassofdifferential operators the most prominent member of which is the Butzer-Wagner derivative. Thegoalofthebookistointroduceresearchersinappliedmathematics,information sciences,computing,andrelatedareastothisparticularconceptindyadicanalysis andvariousgeneralizationsofit. Therationale,andhopefullyanexcuse,forbringingittopublicnotice,isafeel- ingthatduetothepresentsituationininformationsciencesandrelatedtechnology, therecanbe a roomforsome concreteapplicationsof thesedifferentialoperators. Thissituationcanbebrieflysummarizedasfollows. Theinformationandcomputingtechnologiesarecurrentlyfacedwithproblems infurtherexploitingapproacheswhichwereusedinthelastcoupleofdecadesfor developingcomputinghardwarebasedonsemiconductordevicessincetheirphys- ical limits have been nearly reached. There can be observed some similarity with the situationin earlyseventieswhenWalsh dyadicanalysisemergedasan answer todemandsforimprovingperformancesinperformingcertain,primarilysignalpro- cessing related,algorithmsby changingthe underlyingmathematicalfoundations, sincethelimitedcomputingpowerofhardwareavailableatthetimecouldnotpro- videacceptablesolutionsbasedonclassicaltheories. The informationsciences, with this term understoodin a broad sense, are con- tinually challenged to provide answers to demandsfor processing ever increasing amountsofdata.Extractingandprocessingthecontentdatacontainispossibleonly byextensiveuseofcomputinghardwarethatisbeingconstantlyimprovedandper- formance enhanced. Therefore, for both computing technologies and information sciences,thereisa needforessentiallynewdiscoveriespossiblybasedonalterna- tiveapproachestothosecurrentlyused.Thereisagainasimilaritywithintroducing theWalshdyadicanalysisinformertimesasanalternativetoclassicalFourieranal- ysisincertainapplications. Differential operators, as introduced by Newton and Leibniz, can be generally viewedasatoolforestimatingtherateandthedirectionofchangeofsignals.The Gibbsderivativeswereintroducedover45yearsagotoservethesamepurposeinin- formationsciencesastheNewton-Leibnizderivativedoesinmathematicalphysics vii viii Preface andrelatedareas1.Therefore,thesedifferentialoperatorscanbeviewedasanalter- nativetotheexistingdifferentialoperatorstailoredforapplicationinWalshdyadic analysisandextedningtheclassesoffunctionsdifferentiableincertainsense. Atthattime,Walsh (Fourier)analysishadattractedmuchattentionfortworea- sons.First,itenjoyedpeculiarproperties(beingabounded,piecewiseconstantor- thonormalsystem),andsecond,itscoefficientsandtransformscouldbeeasilyand efficiently computed (since it took on only the values ±1). These propertiesgave the Walsh system quite an importantadvantageoverthe classical Fourieranalysis regardingtheavailablecomputerhardware. The Gibbs derivatives were introduced as differential operators having Walsh functions as their eigenfunctions. It might be said that these operators were in- tendedtoprovideamathematicaltoolofferingadvantagesofdifferentialoperators primarilytakingintoaccountdemandsforprocessingofbinaryencodedsignals,but alsothecomputingpowerofferedbythetechnologyatthetime.Therefore,interest in Walsh analysis, in general especially discrete Walsh analysis, and in the Gibbs derivativesin particular, can be explained by two reasons. First engineeringprac- tice requiredfast implementationof variousalgorithmsforsignalandinformation processingbasedonspectral(Walsh-Fourier)analysis,andsecondthelimitedcom- putingpower of the technologyat the time ultimativelydemandedsomethingthat tooklessmemorythanolder,moreestablishedclassicalFourieranalysis. For these reasons, regarding the present situation, we believe that the way of thinkingthatledtothisconceptcanbeequallyusefulindefiningnewconceptsand relatedmethodsincomputingwithapplicationsinsignalandinformationprocessing nowandinfuture. Toserveitsgoal,thebookcontainsreprintsofseveralpaperssettingthefounda- tionsofGibbsdifferentialoperatorsingeneral,andtheButzer-Wagnerderivativein particular,aswellassomegeneralizationsoftheseconcepts. Reprintingoriginalformer publicationsinstead of rewriting the main theory,is motivatedbythefollowingconsiderations. By taking the line of developmentof Walsh analysis as a computationallyless demandingmathematicalfoundationsforsignalprocessingalgorithmsintoaccount, we did not want to rephrase or rewrite the original statements in early papers on this subject for two reasons. First, we believe that the way these statement were presented originally by the authors, has a particular value for the reader. Second, themannerofwritingandthewayofformulatinganddiscussingcertainimportant conceptsoftenreflectsboththefashionofthetimesaswellastheauthorsattitude tothesubject.This,inaway,alsoexpressesothercircumstancesatthetimewhen thenotionswereintroducedandconceptsandtheoriesformulated. Therefore,wedidnotdaredisturbanddestroytheinitialcompositionofthecon- tentsandtheirformulation,asdonebytheauthors,byrewordingthestatementsin presentterminology.Thus,werestrictedthecontributiononourparttotheselection ofwhatweconsidermostinterestingtopresenttothereadersnotice. 1Gibbs,J.E.,“Acontributiontoarevolution?”NPLNews,1971May,1-4,(1971). Itismeanttheinformaticsrevolutionthatstartedataboutthattime(remarkbyeditors) Preface ix Takingthisintoaccount,thereprintedpapersareselectedbythefollowingcrite- ria. Wewantedtoreprintpapersinwhichimportantresultsinthisareawereinitially introducedanddiscussed.Mostofthesepaperswerepresentedatworkshopswhich make them not so easily accessible. That was another motivation to reprint them rather than more elaborated versions that were possibly published latter in some journals.Wealsowantedtopresentpaperswhichsuggestedalternativeapproaches to the Gibbs differentiation,proposedparticular extensionsand generalizationsof theconcept,ordiscussedcertainapplications. The reprinted papers shuld serve as a basis for reading newly written chapters reviewingformerandpresentingrecentandnewresultsinthearea.Thesechapters providereviewsofparticularaspectsofthetheoryofGibbsdifferentiationandare writtenmostlybythefoundersofthistheory.Thebookincludesdiscussionsofre- centdevelopmentsintheareaaswellasacollectionofopenandunsolvedproblems. Attemptingtoprovidearathercompletepictureofthedevelopmentinthefield,we askedthecontributorstothisbookwhostartedtheirresearchinthefieldinseventies andeightiesfortheirreminiscencesofpersonalinvolvementinthesubject. Withahopethatthebookwillserveitspurpose,wethankallthecontributorsas wellasotherfriendswithoutwhosehelpitsappearancewouldhardlybepossible. Acknowledgments The four editors of this monograph,in particular Radomir S. Stankovic´ and Paul L.Butzer,expresstheirthanksforits preparationtothe authorsofthe monograph together with their former students. The initiator of dyadic differentiation, James EdmundGibbs,NationalPhysicalLaboratory,Teddington,UK,diedmuchtooearly, in January 2007, and so could, most unfortunately, not participate in our present project. Mrs.MerionGibbsreportedthataccordingtoherhusbandsdiary,hisfirstdefini- tionofhis”logicalderivative”waswritteninJanuary1967.Ofhis27publications ondyadicderivativessixwerewrittentogetherwithDr.BrianIreland,BathUniver- sity,UK,whojoinedDr.Gibbsintheworkonthissubjectin1971. Thejointworkofauthorstowardsthepresentmonograph,whichtookplaceover aperiodofalmostfouryears,wasveryconstructiveandunbelievablyharmonious, and even during the time from 1969 until 1990 when the chief results of dyadic Walsh analysis were developed by them. This is by no means self-evident since the authors come from ten countries, namely Austria, Canada, China, England, Germany, Hungary, Japan, Russia, Serbia, USA, with quite different mathemati- cal/scientifictraditions. Themajorityofthese,togetherwithsomeoftheirformerstudents,onlymetfor thefirsttimeattheWorkshop”TheoryandApplicationsofGibbsDerivatives”,con- ductedbyRadomirS. Stankovic´ atKupari-Dubrovnik,Yugoslavia,onSeptember 26-28,1989. x Preface AllauthorsexpresstheirspecialthankstoCharlesChui,StanfordUniversity,for acceptingourmonographas Editorof the series ”AtlantisStudies in Mathematics for Engineering and Science” of Atlantis Press, Paris, without any hesitations. It isquitecommonforselectedorcompletepublicationsofoneauthor(whichcould coverseveralfields)toappearinbook-form.Butthisisbynomeanssoforselected papers of all foundingauthors of one field, even together with reader-friendlyre- views of their own papers, to appear in book-form.Further, the authors thank Dr. KeithJonesforhandlingallmattersconcernedwithAtlantisPress,Paris,inanef- ficientandfriendlymanner,aswellasSpringerforproduction,marketing,anddis- tributionofthisbook.Inparticular,thanksareduetoMs.DeviIgnasy,theProject Coordinator,andMs.GajalakshmiSundaram,theProductionEditor,fromSpringer, forexcellentprocessingofthemanuscript,especiallyforverygoodcareofreprinted papers,thesourceversionsofsomeofthemwhichwerenotsoclearandeasilyread- able. TheEditors