Dyadic Walsh Analysis from 1924 Onwards Walsh-Gibbs-Butzer Dyadic Differentiation in Science Volume 1 Foundations

474 Pages·2015·84.292 MB·English

by  Radomir S. Stankovic, Paul L. Butzer, Ferenc Schipp, William R. Wade, Weiyi Su et al.

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
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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