Best approximation in inner product spaces:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
New York [u.a.]
Springer
2001
|
| Schriftenreihe: | CMS books in mathematics
7 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XV, 338 S. graph. Darst. |
| ISBN: | 0387951563 |
Internformat
MARC
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Datensatz im Suchindex
| _version_ | 1804128488490270720 |
|---|---|
| adam_text | xiii
CONTENTS
Preface vii
Acknowledgments xii
Chapter 1. Inner Product Spaces 1
Five Basic Problems 1
Inner Product Spaces 2
Orthogonality 8
Topological Notions 10
Hilbert Space 14
Exercises 15
Historical Notes 19
Chapter 2. Best Approximation 21
Best Approximation 21
Convex Sets 22
Five Basic Problems Revisited 27
Exercises 30
Historical Notes 32
Chapter 3. Existence and Uniqueness of
Best Approximations 33
Existence of Best Approximations 33
Uniqueness of Best Approximations 35
Compactness Concepts 38
Exercises 39
Historical Notes 40
Chapter 4. Characterization of Best Approximations 43
Characterizing Best Approximations 43
Dual Cones 44
Characterizing Best Approximations from Subspaces 50
Gram Schmidt Orthonormalization 51
Fourier Analysis 54
Solutions to the First Three Basic Problems 61
Exercises 64
Historical Notes 69
Chapter 5. The Metric Projection 71
Metric Projections onto Convex Sets 71
Linear Metric Projections 77
xiv CONTENTS
The Reduction Principle 80
Exercises 84
Historical Notes 87
Chapter 6. Bounded Linear Functionate and Best
Approximation from Hyperplanes and Half Spaces 89
Bounded Linear Functionals 89
Representation of Bounded Linear Functionals 93
Best Approximation from Hyperplanes 97
Strong Separation Theorem 102
Best Approximation from Half Spaces 107
Best Approximation from Polyhedra 109
Exercises 117
Historical Notes 122
Chapter 7. Error of Approximation 125
Distance to Convex Sets 125
Distance to Finite Dimensional Subspaces 129
Finite Codimensional Subspaces 133
The Weierstrass Approximation Theorem 139
Miintz s Theorem 143
Exercises 148
Historical Notes 151
Chapter 8. Generalized Solutions of
Linear Equations 155
Linear Operator Equations 155
The Uniform Boundedness and Open Mapping Theorems ... 164
The Closed Range and Bounded Inverse Theorems 168
The Closed Graph Theorem 169
Adjoint of a Linear Operator 171
Generalized Solutions to Operator Equations 177
Generalized Inverse 179
Exercises 187
Historical Notes 191
Chapter 9. The Method of Alternating Projections 193
The Case of Two Subspaces 193
Angle Between Two Subspaces 197
Rate of Convergence for Alternating Projections
(two subspaces) 201
Weak Convergence 203
Dykstra s Algorithm 207
The Case of Affine Sets 215
Rate of Convergence for Alternating Projections 217
Examples 226
Exercises 230
Historical Notes 233
CONTENTS xv
Chapter 10. Constrained Interpolation from a
Convex Set 237
Shape Preserving Interpolation 237
Strong Conical Hull Intersection Property (Strong CHIP) ... 238
Affine Sets 247
Relative Interiors and a Separation Theorem 251
Extremal Subsets of C 261
Constrained Interpolation by Positive Functions 270
Exercises 277
Historical Notes 283
Chapter 11. Interpolation and Approximation 287
Interpolation 287
Simultaneous Approximation and Interpolation 292
Simultaneous Approximation, Interpolation, and
Norm preservation 294
Exercises 295
Historical Notes 298
Chapter 12. Convexity of Chebyshev Sets 301
Is Every Chebyshev Set Convex? 301
Chebyshev Suns 302
Convexity of Boundedly Compact Chebyshev Sets 304
Exercises 306
Historical Notes 307
Appendix 1. Zorn s Lemma 311
Appendix 2. Every Hilbert Space Is e2{I) 312
References 315
Index 331
|
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| dewey-search | 515/.733 21 515/.733 |
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| id | DE-604.BV013676275 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T18:50:02Z |
| institution | BVB |
| isbn | 0387951563 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-009344641 |
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| physical | XV, 338 S. graph. Darst. |
| publishDate | 2001 |
| publishDateSearch | 2001 |
| publishDateSort | 2001 |
| publisher | Springer |
| record_format | marc |
| series | CMS books in mathematics |
| series2 | CMS books in mathematics |
| spelling | Deutsch, Frank Verfasser aut Best approximation in inner product spaces Frank Deutsch New York [u.a.] Springer 2001 XV, 338 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier CMS books in mathematics 7 Approximation, Théorie de l' Espaces à produit scalaire Inner product spaces Approximation theory Innenproduktraum (DE-588)4130366-0 gnd rswk-swf Beste Approximation (DE-588)4144932-0 gnd rswk-swf Approximationstheorie (DE-588)4120913-8 gnd rswk-swf Approximationstheorie (DE-588)4120913-8 s Innenproduktraum (DE-588)4130366-0 s DE-604 Beste Approximation (DE-588)4144932-0 s CMS books in mathematics 7 (DE-604)BV013248581 7 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009344641&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Deutsch, Frank Best approximation in inner product spaces CMS books in mathematics Approximation, Théorie de l' Espaces à produit scalaire Inner product spaces Approximation theory Innenproduktraum (DE-588)4130366-0 gnd Beste Approximation (DE-588)4144932-0 gnd Approximationstheorie (DE-588)4120913-8 gnd |
| subject_GND | (DE-588)4130366-0 (DE-588)4144932-0 (DE-588)4120913-8 |
| title | Best approximation in inner product spaces |
| title_auth | Best approximation in inner product spaces |
| title_exact_search | Best approximation in inner product spaces |
| title_full | Best approximation in inner product spaces Frank Deutsch |
| title_fullStr | Best approximation in inner product spaces Frank Deutsch |
| title_full_unstemmed | Best approximation in inner product spaces Frank Deutsch |
| title_short | Best approximation in inner product spaces |
| title_sort | best approximation in inner product spaces |
| topic | Approximation, Théorie de l' Espaces à produit scalaire Inner product spaces Approximation theory Innenproduktraum (DE-588)4130366-0 gnd Beste Approximation (DE-588)4144932-0 gnd Approximationstheorie (DE-588)4120913-8 gnd |
| topic_facet | Approximation, Théorie de l' Espaces à produit scalaire Inner product spaces Approximation theory Innenproduktraum Beste Approximation Approximationstheorie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009344641&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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