LEADER 05106nam 22006615 450 001 9910136603903321 005 20220407184038.0 024 7 $a10.1007/978-981-10-0530-5 035 $a(CKB)3710000000902993 035 $a(EBL)4718083 035 $a(DE-He213)978-981-10-0530-5 035 $a(MiAaPQ)EBC4718083 035 $a(PPN)196319609 035 $a(EXLCZ)993710000000902993 100 $a20161014d2016 u| 0 101 0 $aeng 135 $aur|n|---||||| 181 $ctxt$2rdacontent 182 $cc$2rdamedia 183 $acr$2rdacarrier 200 10$aTheory of reproducing kernels and applications /$fby Saburou Saitoh, Yoshihiro Sawano 205 $a1st ed. 2016. 210 1$aSingapore :$cSpringer Singapore :$cImprint: Springer,$d2016. 215 $a1 online resource (464 p.) 225 1 $aDevelopments in Mathematics,$x1389-2177 ;$v44 300 $aDescription based upon print version of record. 311 $a981-10-0529-X 311 $a981-10-0530-3 320 $aIncludes bibliographical references and index. 327 $aDefinitions and examples of reproducing kernel Hilbert spaces -- Fundamental properties of RKHS -- Moore Penrose generalized inverses and Tikhonov regularization -- Real inversion formulas of the Laplace transform -- Applications to ordinary differential equations -- Applications to partial differential equations -- Applications to integral equations -- Special topics on reproducing kernels -- Appendices -- Index. 330 $aThis book provides a large extension of the general theory of reproducing kernels published by N. Aronszajn in 1950, with many concrete applications. In Chapter 1, many concrete reproducing kernels are first introduced with detailed information. Chapter 2 presents a general and global theory of reproducing kernels with basic applications in a self-contained way. Many fundamental operations among reproducing kernel Hilbert spaces are dealt with. Chapter 2 is the heart of this book. Chapter 3 is devoted to the Tikhonov regularization using the theory of reproducing kernels with applications to numerical and practical solutions of bounded linear operator equations. In Chapter 4, the numerical real inversion formulas of the Laplace transform are presented by applying the Tikhonov regularization, where the reproducing kernels play a key role in the results. Chapter 5 deals with ordinary differential equations; Chapter 6 includes many concrete results for various fundamental partial differential equations. In Chapter 7, typical integral equations are presented with discretization methods. These chapters are applications of the general theories of Chapter 3 with the purpose of practical and numerical constructions of the solutions. In Chapter 8, hot topics on reproducing kernels are presented; namely, norm inequalities, convolution inequalities, inversion of an arbitrary matrix, representations of inverse mappings, identifications of nonlinear systems, sampling theory, statistical learning theory and membership problems. Relationships among eigen-functions, initial value problems for linear partial differential equations, and reproducing kernels are also presented. Further, new fundamental results on generalized reproducing kernels, generalized delta functions, generalized reproducing kernel Hilbert spaces, and as well, a general integral transform theory are introduced. In three Appendices, the deep theory of Akira Yamada discussing the equality problems in nonlinear norm inequalities, Yamada's unified and generalized inequalities for Opial's inequalities and the concrete and explicit integral representation of the implicit functions are presented. 410 0$aDevelopments in Mathematics,$x1389-2177 ;$v44 606 $aFunctional analysis 606 $aFourier analysis 606 $aFunctions of complex variables 606 $aDifferential equations, Partial 606 $aFunctional Analysis$3https://scigraph.springernature.com/ontologies/product-market-codes/M12066 606 $aFourier Analysis$3https://scigraph.springernature.com/ontologies/product-market-codes/M12058 606 $aFunctions of a Complex Variable$3https://scigraph.springernature.com/ontologies/product-market-codes/M12074 606 $aPartial Differential Equations$3https://scigraph.springernature.com/ontologies/product-market-codes/M12155 615 0$aFunctional analysis. 615 0$aFourier analysis. 615 0$aFunctions of complex variables. 615 0$aDifferential equations, Partial. 615 14$aFunctional Analysis. 615 24$aFourier Analysis. 615 24$aFunctions of a Complex Variable. 615 24$aPartial Differential Equations. 676 $a510 700 $aSaitoh$b Saburou$4aut$4http://id.loc.gov/vocabulary/relators/aut$059776 702 $aSawano$b Yoshihiro$4aut$4http://id.loc.gov/vocabulary/relators/aut 801 0$bMiAaPQ 801 1$bMiAaPQ 801 2$bMiAaPQ 906 $aBOOK 912 $a9910136603903321 996 $aTheory of Reproducing Kernels and Applications$91910223 997 $aUNINA