Topics in Clifford Analysis - Special Volume in Honor of Wolfgang Sprößig PDF
Preview Topics in Clifford Analysis - Special Volume in Honor of Wolfgang Sprößig
Trends in Mathematics Swanhild Bernstein Editor Topics in Clifford Analysis Special Volume in Honor of Wolfgang Sprößig Trends in Mathematics TrendsinMathematicsisaseriesdevotedtothepublicationofvolumesarisingfrom conferences and lecture series focusing on a particular topic from any area of mathematics.Itsaimistomakecurrentdevelopmentsavailabletothecommunityas rapidlyaspossiblewithoutcompromisetoqualityandtoarchivetheseforreference. ProposalsforvolumescanbesubmittedusingtheOnlineBookProjectSubmission Formatourwebsitewww.birkhauser-science.com. Materialsubmittedforpublicationmustbescreenedandpreparedasfollows: Allcontributionsshouldundergoareviewingprocesssimilartothatcarriedoutby journals and be checked for correct use of language which, as a rule, is English. Articles without proofs, or which do not contain any significantly new results, shouldberejected.Highqualitysurveypapers,however,arewelcome. We expecttheorganizerstodelivermanuscriptsina formthatisessentiallyready fordirectreproduction.AnyversionofTEX isacceptable,buttheentirecollection of files must be in one particular dialect of TEX and unified according to simple instructionsavailablefromBirkhäuser. Furthermore, in order to guarantee the timely appearance of the proceedings it is essentialthatthefinalversionoftheentirematerialbesubmittednolaterthanone yearaftertheconference. Moreinformationaboutthisseriesathttp://www.springer.com/series/4961 Swanhild Bernstein Editor Topics in Clifford Analysis Special Volume in Honor of ¨ Wolfgang Sproßig Editor SwanhildBernstein Institutfu¨rAngewandteAnalysis TUBergakademieFreiberg Freiberg,Germany ISSN2297-0215 ISSN2297-024X (electronic) TrendsinMathematics ISBN978-3-030-23853-7 ISBN978-3-030-23854-4 (eBook) https://doi.org/10.1007/978-3-030-23854-4 Mathematics Subject Classification (2010): 30G35, 12D10, 31B05, 32A10, 39A12, 42B20, 51B10, 53C05 ©SpringerNatureSwitzerlandAG2019 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped. Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant protectivelawsandregulationsandthereforefreeforgeneraluse. Thepublisher,theauthors,andtheeditorsaresafetoassumethattheadviceandinformationinthisbook arebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsor theeditorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinorforany errorsoromissionsthatmayhavebeenmade.Thepublisherremainsneutralwithregardtojurisdictional claimsinpublishedmapsandinstitutionalaffiliations. Thisbookispublished under theimprintBirkhäuser, www.birkhauser-science.com, bytheregistered companySpringerNatureSwitzerlandAG. Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland Thisvolumeisdedicated toourcolleague, friendandteacher WolfgangSprößig ontheoccasionofhis 70thbirthday. Hestartedwith quaternionicanalysisinthe 1970sandwas thePhD-advisorof several well-known researchersin thefield of quaternionicandCliffordanalysis. vi W.Sprößig(left)atthe5thInternational Conferenceon CliffordAlgebrasandtheir Applicationsin MathematicalPhysics,1999, inIxtapa,Zihuatanejo(Mexico).1 In therows behindWolfgangaresomeofhiscolleagues andstudents.Second row(fromleft toright): U.Kähler (UniversityofAveiro,PhD-student ofK. Gürlebeck), S. Bernstein(TU BergakademieFreiberg, PhD-studentofW. Sprößig),K. Gürlebeck(Bauhaus-Universität Weimar,PhD-studentand co-authorof W. Sprößig.Last rowinthebackground:S.-L. Eriksson(Universityof Helsinki)). 1WithcourtesyofP.Cerejeiras. Preface This volume, compiled in honour of Wolfgang Sprößig’s 70th birthday, gives an overviewofmodernquaternionicandCliffordanalysis. When Wolfgang Sprößig began his research in the field of quaternionanalysis and elliptic partial differential equations, little was known from multidimensional functiontheory.Back then,functiontheorywas conceivedas a theoryin C or Cn or as a theory of harmonic functions, but not in Rn as a refinement of harmonic functiontheory. It turned out that Clifford algebras are the appropriate tool for refining the harmonicanalysisandfordescribingahigher-dimensionalanalogueoftheCauchy- Riemann system, the so-called generalized Cauchy-Riemann system, and in par- ticular the Dirac operator, which is actually related to the Dirac operator in physics. Today, many completely different topics and theories rely on Clifford analysisasatoolorasaprincipalresearchtopic.Thecontributionstothisvolume exemplify various approaches of Clifford analysis and its application to partial differentialequations,distributions,harmonicanalysisandframeworks,monogenic polynomials,numericalmethods,differentialgeometry,aswellasdiscreteClifford analysis. Iwouldliketothankallthecolleagueswhocontributedtothisvolume—someof thembeinglong-termfriendsofWolfgangSprößig,someofthembeinghisstudents, someof them beinghis friends’disciples, butallof thembeingresearchersin the fieldofCliffordanalysis. Freiberg,Germany SwanhildBernstein March2019 vii Laudation KlausGürlebeckandHelmuthR.Malonek Thiscollection1contains23originalcontributions,submittedby40colleaguesand friends on the occasion of the 70th birthday of Professor Wolfgang Sprößig. The maingoalofthisprefaceistoprovidea—admittedlysubjectiveandincomplete— reviewofthelifeandworkofthecelebrant,andtohighlighthiscontributiontoand hisinfluenceonthedevelopmentofquaternionicandCliffordanalysis,ascientific field with origins in the first half of the twentieth century. The actual importance ofthisfield,anditsvastpotentialforpracticalapplicationsinparticular,havebeen fullyunderstoodinthelate1970sandearly1980s—toaconsiderabledegreedueto theworkofWolfgangSprößig. As the authors of this preface know all too well, Professor Sprößig is not the kind of person to revel in the spotlight. Still, we do hope that this book will give himtheopportunitytolookbackonhislifepathwiththeperspectivethatonlythe passageoftimemaygrantandrecognizesomeofhismanyachievements,reflected here through the work and the eyes of other authors. The picture that emerges is certainlydazzling. WolfgangSprößigwasbornintheSaxoncityofChemnitzonSunday,October 13th,1946,tohislovingparentsDoraandKarlSprößig.From1953until1965,he attendedhighschool,andin1965beganhisstudiesinmathematicsattheUniversity of Technology in Karl-Marx-Stadt (today: Chemnitz). In 1969, he successfully finishedhisstudieswiththe(German)diplomadegree.Atthattime,theUniversity of Technology in Karl-Marx-Stadt was facing big challenges. New departments 1TheauthorsthankMariusMitreaforthelinguisticimprovementofthislaudation. K.Gürlebeck ChairofAppliedMathematics,Bauhaus-UniversitätWeimar,Weimar,Germany e-mail:[email protected] H.R.Malonek DepartamentodeMatemática,UniversidadedeAveiro,Aveiro,Portugal e-mail:[email protected] ix x K.GürlebeckandH.R.Malonek were established in order to meet the high demand caused by the industrial developmentin the region, and the number of students was rapidly increasing. In 1968, a reorganization of all the universities had taken place that granted more freedomto research.Also, increasingattentionwas beingpaid to theapplicability oftheoreticalresultsinpractice.Atthattime,theUniversityofTechnologyinKarl- Marx-Stadthad attracted severalyoungmathematicianswith a “habilitation” who had studied at the universities of Leningrad (today: St. Petersburg) and Moscow, some of whom had also graduated there with a doctoral degree. As a result, the level of activity in pure and applied mathematics increased drastically. Among the new professors who joined this movement was Siegfried Prößdorf, who had graduatedunderthesupervisionofSolomonG.Mikhlin.In1969,WolfgangSprößig begantoworkasayoungassistantinPrößdorf’sresearchgroup.Hecompletedand defendedhisdissertationin1974.Thethesisisentitled“ÜberdieRegularisierung eines Systems zweidimensionaler singulärer Integralgleichungen, dessen Symbol endlich viele Nullstellen ganzzahliger Ordnung besitzt” (English translation: “On theregularizationofasystemoftwo-dimensionalsingularintegralequationswhose symbolhasfinitelymanyzerosofintegerorder”). Havingacquiredagreatdealofprofessionalexpertise,WolfgangSprößigbegan todirecthisintereststothefieldofquaternionicanalysis—which,asweallknow, hasremainedavibrantareaofactivityuntiltoday.Hewrotefiveoriginalpapersthat setthegroundforhishabilitationthesis“EinemehrdimensionaleOperatorenrech- nung über beschränkten Gebieten des Euklidischen Raumes und ihre Anwendung aufdieLösungvonGleichungen”(Englishtranslation:“Amultidimensionalopera- torcalculusonboundeddomainsofEuclideanspace,anditsapplicationtosolving equations”), published in 1979. These first five papers as well as the habilitation thesis were the beginning of an ambitious research program, which would take Wolfgang many decades to complete. But step by step, it eventually became a reality—aprocessthathasstretchedthroughthe entire academiclife of Wolfgang Sprößig.Envisioningsomethingahead of one’stime is impressive;but havingthe abilityandtenacitytomakeitrealistrulymomentous.WolfgangSprößig’soriginal insightisnowpartoftheuniversalmathematicallandscape. In the early stages, quaternionic analysis had belonged to the fringes of math- ematics and was regardedas merely esoteric. Wolfgang Sprößig challenged these narrow perceptions and succeeded in changing the status quo of quaternionic analysis. With his calm yet persuasive demeanour,he made a compelling case by highlightingboththetheoreticalimportanceofthisfieldaswellasitspotentialfor practicalapplicability. WolfgangSprößighasaninnatetalentforexplainingcomplextopics.Thismade himaperspicuousteacher,asalreadyhisfirststudentscouldexperience.Hislecture onpartialdifferentialequationsin1977wasaremarkableexample.Inthislecture, hehadincorporatedsomeofhisownoriginalresearchresultsthatwerenewatthe time. This gave the audience a chance to acquaint themselves in a most effective manner with quaternion-valuedfunctions, spatial Cauchy-Riemann equations, the generalized Cauchy integral, and last but not least the famous T-operator. This type of lectures was both a revelation and a challenge for the involved students,
The list of books you might like
