Theory of reproducing kernels and applications
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[2016]
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| Schriftenreihe: | Developments in Mathematics
volume 44 |
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| 245 | 1 | 0 | |a Theory of reproducing kernels and applications |c Saburou Saitoh, Yoshihiro Sawano |
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| 264 | 4 | |c © 2016 | |
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| 650 | 4 | |a Partial differential equations | |
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| 650 | 4 | |a Fourier Analysis | |
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Datensatz im Suchindex
| _version_ | 1819617749673967616 |
|---|---|
| adam_text | Contents 1 Deflnitions and Examples of Reproducing Kernel Hilbert Spaces...... 1.1 What Is an RKHS?............................................................................... 1.1.1 Definition................................................................................. 1.1.2 Orientation of Chap. 1............................................................ 1.2 Paley Wiener Reproducing Kernels ................................................... 1.2.1 Paley Wiener Space................................................................. 1.2.2 A Characterization Using the Fourier Transform................ 1.3 RKHS of Sobolev l pe....................................................................... 1.3.1 Weak-Derivatives and Sobolev Spaces ................................. 1.3.2 1-Dimensional Case............................................................... 1.3.3 In Connection with 1-Dimensional Wave Equations............ 1.3.4 Higher Regularity RKHS....................................................... 1.3.5 Higher-Dimensional Case...................................................... 1.4 RKHS and Complex Analysis on C................................................... 1.4.1 RKHS on 4(1)........................................................................ 1.4.2 RKHS on C............................................................................ 1.4.3 RKHS on C 4(1)................................................................ 1.4.4 RKHS on a Small Neighborhood of the Origin................... 1.4.5 Bergman Kernel on
4(1)....................................................... 1.4.6 Bergman Selberg Reproducing Kernel ................................. 1.4.7 Pullback of Bergman Spaces................................................. 1.5 RKHS in the Spaces of Polynomials.................................................. 1.5.1 General Properties of Orthonormal Systems........................ 1.5.2 Examples of Orthonormal Systems....................................... 1.5.3 Polynomial Reproducing Kernel Hilbert Spaces................. 1.5.4 RKHS Constructed by Meixner-Type Polynomials............. 1.5.5 RKHS for Trigonometric Polynomials ................................ 1.6 Graphs and Reproducing Kernels........................................................ 1.6.1 RKHSJ%............................................................................... 1.6.2 Gram Matrices........................................................................ 1 1 1 2 3 3 6 8 8 10 18 21 22 24 25 29 31 34 35 37 39 41 41 46 49 51 52 53 54 58 xi
xii Contents 1.7 2 Green’s Functions and Reproducing Kernels ..................................... 1.7.1 Basic Concepts of Dirac’s Delta Function, Green’s Functions and Reproducing Kernels....................... 1.7.2 Relationship Between Dirac’s Delta Function, Green’s Functions and Reproducing Kernels......... 61 1.7.3 The Simplest Example............................................................ Fundamental Properties of RKHS............................................................. 2.1 Basic Properties of RKHS.................................................................... 2.1.1 Elementary Properties of RKHS............................................ 2.1.2 Positive Definite Quadratic form Functions.......................... 2.1.3 Smoothness of Functions and Kernels of RKHS.................. 2.2 Operations of One Reproducing Kernel Hilbert Spaces.................... 2.2.1 Restriction of a Reproducing Kernel...................................... 2.2.2 Pullback of a Reproducing Kernel by Any Mapping ........... 2.2.3 Balloon of Reproducing Kernel Hilbert Spaces.................... 2.2.4 Squeezing of RKHS................................................................ 2.2.5 Transforms of RKHS by Operators........................................ 2.2.6 Pullback of Reproducing Kernels by a Hilbert Space-Valued Function.......................................................... 2.3 Operations of More Than One Reproducing Kernel Hilbert Space.............................................................................. 93 2.3.1 Sum of Reproducing Kernels
................................................. 2.3.2 Sum of Arbitrary Abstract Hilbert Spaces............................. 2.3.3 Pasting of Reproducing Kernel Hilbert Spaces..................... 2.3.4 Tensor Product of Hilbert Spaces........................................... 2.3.5 Increasing Sequences of RKHS................................................ 2.3.6 Wedge Product of RKHS.......................................................... 2.4 Construction of Reproducing Kernel Hilbert Spaces......................... 2.4.1 Use Linear Mappings and CONS......................................... 2.4.2 Use the Fourier Integral......................................................... 2.4.3 Mercer Expansion: A General Approach.............................. 2.4.4 Realization by Discrete Points............................................... 2.4.5 Aveiro Discretization Method............................................... 2.5 RKHS and Linear Mappings................................................................ 2.5.1 Identification of the Images of Linear Transforms in Terms of the Adjoint ...................... 134 2.5.2 Identification of the Image of a Linear Mapping.................. 2.5.3 Image Identification by Reproducing Kernels...................... 2.5.4 Integra-Differential Equations.............................................. 2.5.5 Inversion Mapping from Many Types of Information Data...................................................... 148 2.5.6 Random Fields Estimations................................................... 2.5.7 Support Vector Machines and
Probability Theory for Data Analysis....................................................... 60 60 62 65 65 65 67 73 78 78 81 82 86 89 92 93 100 103 105 109 113 115 115 119 120 125 131 133 134 141 147 150 151
xiii Contents 3 2.5.8 Inverse Formulas Using CONS ............................................. 2.5.9 A General Fundamental Theorem on Inversions.................. 2.5.10 Nonharmonic Transforms..................................................... 2.5.11 Determinations of Linear Systems........................................ 153 154 158 159 Moore Penrose Generalized Inverses and Tikhonov Regularization... 161 3.1 3.2 3.3 4 Real Inversion Formulas of the Laplace Transform................................ 4.1 5 The Best Approximations and the Moore Penrose Generalized Inverses................................................................. 161 3.1.1 Frames...................................................................................... 3.1.2 A General Example of the Moore Penrose Inverses ............ 3.1.3 Best Approximation Problems............................................... 3.1.4 Applications to Best Approximation Problems................... 3.1.5 Applications to Operator Equations with a Parameter........ Spectral Analysis and Tikhonov Regularization................................ 3.2.1 Spectral Analysis.................................................................... 3.2.2 Tikhonov Regularization and Reproducing Kernels........... 3.2.3 Representations of the Solutions of the Tikhonov Functional Equation............................... 3.2.4 Applications of Tikhonov Regularization: Approximations by Sobolev Spaces...................... General Fractional Functions............................................................... 3.3.1 What Is a Fractional
Function?............................................... 3.3.2 An Approach by Using RKHSs............................................. Real Inversion Formulas of the Laplace Transform .......................... 4.1.1 Problem and Orientation ....................................................... 4.1.2 Known Real Inversion Formulas of the Laplace Transform................................................................ 198 4.1.3 Compactness of the Modified LaplaceTransform............... 4.1.4 Examples of the Weights........................................................ 4.1.5 Real Inversion of the Laplace Transform Through Singular ValuesDecomposition............... 208 4.1.6 Real Inversion of the Laplace Transform Using Tikhonov Regularization........................................ 4.1.7 Optimal Real Inversion Formulas Using a Finite Number of Data....................................................... 212 Applications to Ordinary Differential Equations...................................... 5.1 Use Tikhonov Regularization........................................................... 5.1.1 Ordinary Linear Differential Equations with Constant Coefficients............................................................. 5.1.2 Variable Coefficients Case .................................................... 5.1.3 Finite Interval Cases............................................................... 5.1.4 Exact Algorithm..................................................................... 5.1.5 One Point Boundary Condition Case................................... 161 165 166 169 174 179
179 184 186 190 193 193 194 197 197 197 200 205 209 217 217 217 220 221 222 223
xiv Contents 5.2 6 Applications to Partial Differential Equations........................................... 6.1 6.2 6.3 6.4 6.5 7 Discrete Ordinary Linear Differential Equations............................... 224 5.2.1 Use a Finite Number of Data.................................................. 224 5.2.2 Discrete Inverse Source Problems......................................... 229 Poisson’s Equation............................................................................... 6.1.1 Construction of the Solutions................................................ 6.1.2 Source Inversions in the Poisson Equation.......................... 6.1.3 Constructing Solutions Satisfying Boundary Conditions ... Laplace’s Equation............................................................................... 6.2.1 Numerical Dirichlet Problems.............................................. 6.2.2 Algorithm by the Iteration Method ...................................... 6.2.3 The Cauchy Problem.............................................................. 6.2.4 Use a Finite Number of Data for the Cauchy Problem......... 6.2.5 Error Estimates......................................................................... Heat Equation........................................................................................ 6.3.1 Applications to the Inversion to 1-Dimensional Heat Conduction...................................................... 253 6.3.2 Use Sobolev Spaces ................................................................ 6.3.3 Use of Paley Wiener Spaces (Sine
Methods)........................ 6.3.4 Source Inversions of Heat Conduction Using a Finite Number of Data............................................. 262 6.3.5 Natural Outputs and Global Inputs of Linear Systems........ Wave Equation....................................................................................... 6.4.1 Inverse Problems in the 1-Dimensional Wave Equation .... 6.4.2 Inverse Problems Using a Finite Number of Data................ General Inhomogeneous PDEs on Whole Spaces ............................. 6.5.1 Approximate Solutions by Tikhonov Regularization.......... 6.5.2 Convergence Properties of the Approximate Solutions....... 6.5.3 General Discrete Partial Differential Equations and Inverse Formulas............................................................. 6.5.4 PDEs and Inverse Problems: A General Approach ............ Applications to Integral Equations................................................................ 7.1 Singular Integral Equations................................................................ 7.1.1 Mathematical Formulation and an Approach via the Paley Wiener Spaces......................................... 7.1.2 The Complex Constant Coefficients Case on the Whole Line................................................... 297 7.1.3 Construction of Approximate Solutions by Using a CONS Expansion..................................................... 7.1.4 General Singular Integral Equations...................................... 7.1.5 Discrete Singular Integral Equations ..................................... 7.1.6 Inverse Source Problems
by a Finite Number of Data........ 231 231 231 238 243 245 245 247 249 250 252 253 255 257 269 273 273 282 284 284 287 289 293 295 295 295 303 306 308 311
xv Contents 7.2 8 Convolution Integral Equations............................................................ 7.2.1 Example of Convolution Integral Equations......................... 7.2.2 An Approach Using RKHSs................................................. 7.2.3 Integral Equations with Mixed Toeplitz Hankel Kernel .... 313 313 313 315 Special Topics on Reproducing Kernels.................................................... 321 8.1 Norm Inequalities .................................................................................. 321 8.1.1 The Bergman Norm and the Szegö Norm: An Overview... 321 8.1.2 Examples of Norm Inequalities via RHKSs................... 323 8.1.3 General Nonlinear Transforms of Reproducing Kernels.... 325 8.1.4 Convolution Norm Inequalities............................................ 330 8.2 Inequalities for Gram Matrices............................................................. 334 8.3 Inversions................................................................................................ 337 8.3.1 Inversion for Any Matrix by Tikhonov Regularization. 337 8.3.2 Representation of Inverse Functions............................... 340 8.3.3 Identifications of NonlinearSystems.................................... 346 8.4 Sampling Theory................................................................................... 347 8.4.1 The Information: Loss Error in Sampling Theory......... 350 8.4.2 A General Kramer-Type Lemma in Sampling Theory.. 354 8.5 Error and Convergence Rate Estimates in Statistical Learning
Theory......................................................................... 357 8.6 Membership Problems for RKHSs....................................................... 363 8.6.1 How to Obtain Smoothing Properties and Analyticity of Functions UsingComputers?.......... 363 8.6.2 Band Preserving, Phase Retrieval and Related Problems ... 365 8.7 Eigenfunctions, Initial Value Problems,and Reproducing Kernels.. 366 8.7.1 Formulation of the Problem.................................................. 366 8.7.2 The Derivative Operator and the Corresponding Exponential Function Case..................................... 370 8.7.3 The Case of the Operator L(D) = D2................................... 372 8.7.4 How to Obtain Eigenfunctions.............................................. 373 8.8 Generalized Reproducing Kernels and Generalized Delta Functions..................................................................................... 378 8.8.1 What Is a Reproducing Kernel?............................................ 378 8.8.2 A Generalized Reproducing Kernel...................................... 379 8.8.3 Generalized Reproducing Kernels........................................ 380 8.9 General Integral Transforms by the Concept of Generalized Reproducing Kernels........................................ 381 8.9.1 Formulation of a Fundamental Problem............................... 381 8.9.2 Completion Property ............................................................. 383 8.9.3 Convergence of/((л) = (F, hx)L2(ItXim) for F e L2(7, dm)... 384 8.9.4 Inversion of Integral
Transforms........................................... 384 8.9.5 Discrete Versions................................................................... 385
xvi Contents Appendices............................................................................................................ 387 A.l Equality Problems for Norm Inequalities........................................... 387 A. 1.1 Introduction of the Results by Akira Yamada....................... 387 A. 1.2 Equality Conditions for the Norm Inequalities..................... 395 A. 1.3 The Case of Polynomial Ring................................................. 399 A. 1.4 Algebra of Meromorphic Functions ...................................... 400 A. 1.5 Applications............................................................................. 403 A.2 Generalizations of Opial’s Inequality.................................................. 406 A.2.1 Main Inequality....................................................................... 407 A.2.2 Equality Condition................................................................... 409 A.2.3 Applications............................................................................. 411 A.3 Explicit Integral Representations of Implicit Functions.................... 413 A.3.1 2-Dimensional Case................................................................ 413 A.3.2 Representations of Implicit Functions................................... 415 A.3.3 The 2-Dimensional Case: n = I,к = 1 ................................ 419 A.3.4 The 3-Dimensional Cases: n = I, к = 2 or n= 2,к = 1 .. 421 A.4 Overview on the Theory of Reproducing Kernels............................. 422
References............................................................................................................. 425 Index...................................................................................................................... 447
|
| any_adam_object | 1 |
| author | Saitoh, Saburou 1944- Sawano, Yoshihiro |
| author_GND | (DE-588)107249583X (DE-588)1044114002 |
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| dewey-ones | 515 - Analysis |
| dewey-raw | 515.7 |
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| dewey-sort | 3515.7 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| id | DE-604.BV043877190 |
| illustrated | Not Illustrated |
| indexdate | 2024-12-24T05:27:11Z |
| institution | BVB |
| isbn | 9789811005299 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-029286884 |
| oclc_num | 966299926 |
| open_access_boolean | |
| owner | DE-188 DE-11 DE-473 DE-BY-UBG |
| owner_facet | DE-188 DE-11 DE-473 DE-BY-UBG |
| physical | xviii, 452 Seiten |
| publishDate | 2016 |
| publishDateSearch | 2016 |
| publishDateSort | 2016 |
| publisher | Springer |
| record_format | marc |
| series | Developments in Mathematics |
| series2 | Developments in Mathematics |
| spellingShingle | Saitoh, Saburou 1944- Sawano, Yoshihiro Theory of reproducing kernels and applications Developments in Mathematics Mathematics Fourier analysis Functional analysis Functions of complex variables Partial differential equations Functional Analysis Fourier Analysis Functions of a Complex Variable Partial Differential Equations Mathematik Kernfunktion (DE-588)4163607-7 gnd Hilbert-Raum (DE-588)4159850-7 gnd Funktionalanalysis (DE-588)4018916-8 gnd |
| subject_GND | (DE-588)4163607-7 (DE-588)4159850-7 (DE-588)4018916-8 |
| title | Theory of reproducing kernels and applications |
| title_auth | Theory of reproducing kernels and applications |
| title_exact_search | Theory of reproducing kernels and applications |
| title_full | Theory of reproducing kernels and applications Saburou Saitoh, Yoshihiro Sawano |
| title_fullStr | Theory of reproducing kernels and applications Saburou Saitoh, Yoshihiro Sawano |
| title_full_unstemmed | Theory of reproducing kernels and applications Saburou Saitoh, Yoshihiro Sawano |
| title_short | Theory of reproducing kernels and applications |
| title_sort | theory of reproducing kernels and applications |
| topic | Mathematics Fourier analysis Functional analysis Functions of complex variables Partial differential equations Functional Analysis Fourier Analysis Functions of a Complex Variable Partial Differential Equations Mathematik Kernfunktion (DE-588)4163607-7 gnd Hilbert-Raum (DE-588)4159850-7 gnd Funktionalanalysis (DE-588)4018916-8 gnd |
| topic_facet | Mathematics Fourier analysis Functional analysis Functions of complex variables Partial differential equations Functional Analysis Fourier Analysis Functions of a Complex Variable Partial Differential Equations Mathematik Kernfunktion Hilbert-Raum Funktionalanalysis |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=029286884&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV013103064 |
| work_keys_str_mv | AT saitohsaburou theoryofreproducingkernelsandapplications AT sawanoyoshihiro theoryofreproducingkernelsandapplications |