Theory of Reproducing Kernels and Applications PDF
Preview Theory of Reproducing Kernels and Applications
Developments in Mathematics Saburou Saitoh Yoshihiro Sawano Theory of Reproducing Kernels and Applications Developments in Mathematics Volume44 SeriesEditors: KrishnaswamiAlladi,UniversityofFlorida,Gainesville,FL,USA HershelM.Farkas,HebrewUniversityofJerusalem,Jerusalem,Israel Moreinformationaboutthisseriesathttp://www.springer.com/series/5834 Saburou Saitoh • Yoshihiro Sawano Theory of Reproducing Kernels and Applications 123 SaburouSaitoh YoshihiroSawano ProfessorEmeritus DepartmentofMathematics GunmaUniversity andInformationScience Kiryu,Gunma,Japan TokyoMetropolitanUniversity Hachioji,Tokyo,Japan ISSN1389-2177 ISSN2197-795X (electronic) DevelopmentsinMathematics ISBN978-981-10-0529-9 ISBN978-981-10-0530-5 (eBook) DOI10.1007/978-981-10-0530-5 LibraryofCongressControlNumber:2016951457 ©SpringerScience+BusinessMediaSingapore2016 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,theauthorsandtheeditorsaresafetoassumethattheadviceandinformationinthisbook arebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsor theeditorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinorforany errorsoromissionsthatmayhavebeenmade. Printedonacid-freepaper ThisSpringerimprintispublishedbySpringerNature TheregisteredcompanyisSpringerScience+BusinessMediaSingaporePteLtd. Preface The theory of reproducing kernels started with two papers of 1921 [449] and 1922 [45] which dealt with typical reproducing kernels of Szegö and Bergman, and since then the theory has been developed into a large and deep theory in complexanalysisbymanymathematicians.However,precisely,reproducingkernels appearedpreviouslyduringthefirstdecadeofthetwentiethcenturybyS.Zaremba [497] in his work on boundary value problems for harmonic and biharmonic functions. But he did not develop any further theory for the reproducing property. Furthermore, in fact, we know many concrete reproducing kernels for spaces of polynomials and trigonometric functions from much older days, as we will see in thisbook.Meanwhile,thegeneraltheoryofreproducingkernelswasestablishedina completeformbyN.Aronszajn[28]in1950.Furthermore,L.Schwartz[428],who is a Fields medalist and founded distribution theory, developed the general theory remarkablyin1964withapaperofover140pages. Thegeneraltheoryiscertainlybeautiful.Itseems,however,thatforalongtime we have overlooked the importance of the general theory of reproducing kernels. Wewerenotabletofindanessentialreasonwhythetheoryisimportant.Indeed,it wasanabstracttheory,andfromthetheory,wewerenotabletoderiveanydefinite resultsandanyessentialdevelopmentsinmathematics.ThetheorybySchwartzis great;however,itsimportanceremainedunnoticedforalongtime:Itisstillignored. WhenweconsiderlinearmappingsintheframeworkofHilbertspaces,wewill encounter in a natural way the concept of reproducing kernels; then the general theoryisnotrestrictedtoBergmanandSzegökernels,butthegeneraltheoryisas importantastheconceptofHilbertspaces.Itisafundamentalconceptandimportant mathematics. The general theory of reproducing kernels is based on elementary theorems on Hilbert spaces. The theory of Hilbert spaces is the minimum core of functional analysis. However, when the general theory is combined with linear mappings on Hilbert spaces, it will have many relations in various fields, and its fruitful applications will spread over to differential equations, integral equations, generalizations of the Pythagorean theorem, inverse problems, sampling theory, nonlineartransformsinconnectionwithlinearmappings,variousoperatorsamong v vi Preface Hilbert spaces, and many other broad fields. Furthermore, when we apply the general theory of reproducing kernels to the Tikhonov regularization, it produces approximatesolutionsforequationsonHilbertspaceswhichcontainboundedlinear operators.Lookingfromthepointofviewofcomputerusersatnumericalsolutions, wewillseethattheyarefundamentalandhavepracticalapplications. Concrete reproducing kernels like Bergman and Szegö kernels will produce many wide and broad results in complex analysis. They developed some deep theoryandleadtoprofoundresultsincomplexanalysiscontainingseveralcomplex variables. Meanwhile, the formal general theory by Aronszajn also has favorable connections with various fields like learning theory, support vector machines, stochastictheory,andoperatortheoryonHilbertspaces. In this book, we will concentrate on the general theory of reproducing kernels developed by Aronszajn while keeping in mind the theory combined with linear mappings and applications of the general theory to the Tikhonov regularization. We will present many concrete applications from the point of view of numerical solutionsforcomputeruse.Thesetopicswillbegeneralandfundamentalformany mathematicalscientistsbeyondmathematiciansasincalculusandlinearalgebrain anundergraduatecourse. One of our strong motivations for writing this book was provided by the historicalsuccessofnumericalandrealinversionformulasoftheLaplacetransform, which is a famous ill-posed and difficult problem, and, in fact, we will give their mathematical theory and formulas, as clear evidence of the definite power of the theory of reproducing kernels by combining the Tikhonov regularization. For the algorithmbasedonthetheory,HiroshiFujiwaramadethesoftwareandwecanuse itthroughhishelpfulguide. Theweb[159]isanopensourcetohisinverseLaplacetransform. For these topics, we will need background materials like integration theory, fundamentalHilbertspacetheory,theFouriertransform,andtheLaplacetransform. InChap.1,wewillgivemanyconcretereproducingkernelsfirst,andinChap.2, we develop the general theory of reproducing kernels with general and broad applicationsbycombiningitwithlinearmappings. In Chap.3, we will apply the general and global theory of reproducing kernels to the Tikhonov regularization in a lucid manner. We stand on the point of view of numerical solutions of bounded linear operator equations on Hilbert spaces for computeruseinadefiniteandself-containedway. Chapter 4 is intended as an introduction to what Hiroshi Fujiwara did. In particular,Fujiwarasolvedlinearsimultaneousequationswith6,000unknownsby means of discretization of a Fredholm integral equation of the second kind. This integral equation of the second kind was derived by the Tikhonov regularization and the reproducing kernel method in the above real inversion formula. At this moment, theoretically we will use all the data of the output—in fact, 6,000 pieces of data. Fujiwara gave solutions in 600 digits precision with the data of 10GB forsolutions.Thisfacthadagreatimpactontheauthors.Computerpowerandits algorithmswillimproveyearbyyear.Meanwhile,wecanpracticallyobtainafinite amountofobservationdata,andsoweexpecttoobtainsolutionsintermsofafinite Preface vii number of data for various forward and inverse problems. Thanks to the power of computers,wewillbeabletorealizemoredirectandsimplealgorithms,andsowe haveincludedresultsbasedonafiniteamountofobservationdata.Thismethodwill giveanewdiscretizationprinciple. Chapter 5 deals with the applications to ordinary differential equations such as fundamentalequationsy00C˛y0CˇyD0,where˛andˇcanbegeneralfunctions. Sometimes,weconsiderthecasewhentheboundaryconditioncomesintoplay. As one main substance of new results, in Chap.6, we present many concrete results for various fundamental partial differential equations. Here we take up the Poissonequation,theLaplaceequation,theheatequation,andthewaveequation. Similarly, in Chap.7, we deal with integral equations. We will consider typical singular integral equations, convolution equations, convolution integral equations, andintegralequationswiththemixedToeplitzandHankelkernel. In Chap.8, we refer to specially hot topics and important materials on repro- ducing kernels, namely, norm inequalities, convolution inequalities, inversion of anarbitrarymatrix,representationofinversemappings,identificationofnonlinear systems, sampling theory, statistical learning theory, and membership problems. Thiswillyieldanewmethodofhowtocatchanalyticityandsmoothingproperties of functions by computers. Furthermore, we will see basic relationships among eigenfunctions, initial value problems for linear partial differential equations, and reproducing kernels, and we will refer to a new type of general sampling theory withnumericalexperiments.Inthelasttwosubsections,weaddednewfundamental results on generalized reproducing kernels, generalized delta functions, general- ized reproducing kernel Hilbert spaces, and general integral transform theory. In particular, any separable Hilbert space consisting of functions may be viewed as generalized reproducing kernel Hilbert spaces, and the general integral transform theorymaybeextendedtoageneralframework. Finally,anappendixisprovided.InSect.A.1,weintroducethetheoryofAkira Yamadadiscussingequalityproblemsinnonlinearnorminequalitiesinreproducing kernel Hilbert spaces; indeed, we may be surprised at his general theory of reproducing kernels. In Sect.A.2, we introduce Yamadafs unified and generalized inequalities for Opialfs inequalities. Similar but different generalizations were independently published by Nguyen Du Vi Nhan, Dinh Thanh Duc, and Vu Kim Tuan,inthesameyear.InSect.A.3,weintroduceconcreteintegralrepresentations of implicit functions. We rely upon the implicit function theory guaranteeing the existence of implicit functions. The fundamental result was obtained as a great developmentofageneralabstracttheoryofreproducingkernels. Kiryu,Japan SaburouSaitoh Hachioji,Japan YoshihiroSawano November2015 Acknowledgments Theauthorsthankthefollowingauthorsofthetextbooks: 1. AlainBerlinetandChristineThomas-Agnan:ReproducingKernelHilbertSpaces inProbabilityandStatistics 2. BaverOkutmusturandAurelianGheondea:ReproducingKernelHilbertSpaces: TheBasics,BergmanSpaces,andInterpolationProblemsonreproducingkernels thatwereveryinstructiveforourbook ProfessorH.G.W.Begehrencouragedthepublicationofthisbook. The three referees gave valuable comments and suggestions to the first draft of thisbook. Thefollowingmathematicianskindlysenttheirpapersortheirtextfilesorkind suggestionsforourbookpublication: Luis Daniel Abreu, Kaname Amano, Joseph A. Ball, P. L. Butzer, L.P. Castro, MinggenCui,HiroshiFujiwara,AntonioG.Garcia,J.R.Higgins,HiromichiItou, M. T. Garayev, Kenji Fukumizu, Tsutomu Matsuura, Yan Mo, J. Morais, Nguyen DuViNhan,MasaharuNishio,TakeoOhsawa,HidemitsuOgawa,TaoQian,A.G. Ramm,M.M.Rodrigues,MichioSeto,FethiSoltani,N.S.Stylianopoulos,Mariko Takagi,AkiraYamada,MasatoYamada,HiroyukiYamagishi,NguyenMinhTuan, VuKimTuan,MasahiroYukawa,andKohtaroWatanabe. This work of the first author was supported in part by Portuguese funds through the CIDMA (Center for Research and Development in Mathematics and Applications) and the Portuguese Foundation for Science and Technology (FCT), withinprojectPEst-OE/MAT/UI4106/2014. ThefirstauthorwasalsosupportedinpartbytheGrant-in-AidfortheScientific Research(C)(2)(No.21540111,24540113)fromtheJapanSocietyforthePromo- tionofScience,andthesecondauthorwassupportedbytheGrant-in-AidforYoung Scientists(B)(No.21740104,24740085)fromtheJapanSocietyforthePromotion ofScience. ix Contents 1 DefinitionsandExamplesofReproducingKernelHilbertSpaces ..... 1 1.1 WhatIsanRKHS? ..................................................... 1 1.1.1 Definition...................................................... 1 1.1.2 OrientationofChap.1........................................ 2 1.2 PaleyWienerReproducingKernels ................................... 3 1.2.1 PaleyWienerSpace........................................... 3 1.2.2 ACharacterizationUsingtheFourierTransform ........... 6 1.3 RKHSofSobolevType ................................................ 8 1.3.1 Weak-DerivativesandSobolevSpaces ...................... 8 1.3.2 1-DimensionalCase .......................................... 10 1.3.3 InConnectionwith1-DimensionalWaveEquations........ 18 1.3.4 HigherRegularityRKHS..................................... 21 1.3.5 Higher-DimensionalCase.................................... 22 1.4 RKHSandComplexAnalysisonC................................... 24 1.4.1 RKHSon(cid:2).1/................................................ 25 1.4.2 RKHSonC ................................................... 29 1.4.3 RKHSonCn(cid:2).1/........................................... 31 1.4.4 RKHSonaSmallNeighborhoodoftheOrigin ............. 34 1.4.5 BergmanKernelon(cid:2).1/..................................... 35 1.4.6 BergmanSelbergReproducingKernel ...................... 37 1.4.7 PullbackofBergmanSpaces................................. 39 1.5 RKHSintheSpacesofPolynomials .................................. 41 1.5.1 GeneralPropertiesofOrthonormalSystems ................ 41 1.5.2 ExamplesofOrthonormalSystems.......................... 46 1.5.3 PolynomialReproducingKernelHilbertSpaces............ 49 1.5.4 RKHSConstructedbyMeixner-TypePolynomials......... 51 1.5.5 RKHSforTrigonometricPolynomials ...................... 52 1.6 GraphsandReproducingKernels...................................... 53 1.6.1 RKHSH ..................................................... 54 G 1.6.2 GramMatrices................................................ 58 xi
Description:The list of books you might like
