Variational theory of splines:
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| Hauptverfasser: | , |
|---|---|
| Format: | Buch |
| Sprache: | Undetermined |
| Veröffentlicht: |
New York [u.a.]
Kluwer Acad./Plenum Publ.
2001
|
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XVII, 280 S. |
| ISBN: | 0306466422 |
Internformat
MARC
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| 100 | 1 | |a Bezaev, Anatolij Ju. |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Variational theory of splines |c Anatoly Yu. Bezhaev and Vladimir A. Vasilenko |
| 264 | 1 | |a New York [u.a.] |b Kluwer Acad./Plenum Publ. |c 2001 | |
| 300 | |a XVII, 280 S. | ||
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| 337 | |b n |2 rdamedia | ||
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| 700 | 1 | |a Vasilenko, Vladimir A. |e Verfasser |4 aut | |
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Datensatz im Suchindex
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|---|---|
| adam_text | IMAGE 1
VARIATIONAL THEORY
OF SPLINES
ANATOLY YU. BEZHAEV AND
VLADIMIR A. VASILENKO INSTITUTE OF COMPUTATIONAL MATHEMATICS AND
MATHEMATICAL GEOPHYSICS NOVOSIBIRSK, RUSSIA
KLUWER ACADEMIC/PLENUM PUBLISHERS
NEW YORK, BOSTON, DORDRECHT, LONDON, MOSCOW
IMAGE 2
CONTENTS
PREFACE XI
INTRODUCTION: A GUIDE TO THE READER XIII
1. SPLINES IN HILBERT SPACES 1
1.1 INTERPOLATING, SMOOTHING, AND MIXED SPLINES 1
1.1.1 MAIN DEFINITIONS 1
1.1.2 INTERPOLATION 2
1.1.3 SMOOTHING 4
1.1.4 MIXED SPLINES 5
1.1.5 FUNCTIONAL EQUATIONS ON SPLINES 7
1.1.6 PSEUDO-INTERPOLATING SPLINES 9
1.1.7 ANY SMOOTHING SPLINE IS AN INTERPOLATING ONE 9
1.2 SPLINES AND EQUIVALENT NORMS IN HUBERT SPACES . . .. 10
1.2.1 MAIN THEOREM 10
1.2.2 EXAMPLES OF EQUIVALENT NORMS IN SOBOLEV SPACES 11 1.3 EXAMPLES OF
SPLINES 14
1.3.1 ONE-DIMENSIONAL SPLINES BY POINT EVALUATIONS . 14 1.3.2
ONE-DIMENSIONAL SPLINES BY LOCAL INTEGRALS . .. 15
1.3.3 MULTI-DIMENSIONAL M -SPLINES BY POINT EVALUA TIONS 16
1.3.4 M -SPLINES BY LOCAL INTEGRALS 16
1.3.5 FINITE-DIMENSIONAL M -SPLINES 17
1.4 STRUCTURE OF SPLINE PROJECTORS 18
1.4.1 MAXIMAL SPLINE-PAIRS 18
1.4.2 INTERPOLATING SPLINE-PROJECTOR 19
1.4.3 SMOOTHING SPLINE-OPERATOR 21
2. REPRODUCING MAPPINGS AND CHARACTERIZATION OF SPLINES 23
2.1 REPRODUCING MAPPINGS AND KERNELS 24
2.1.1 DEFINITIONS 24
V
IMAGE 3
VI
VARIATION R AL THEORY OF SPLINES
2.1.2 BASIC PROPERTIES OF REPRODUCING MAPPINGS . .. 26
2.1.3 BASIC PROPERTIES OF REPRODUCING KERNELS . . .. 28
2.1.4 ADDITIONAL PROPERTIES OF REPRODUCING MAPPINGS AND KERNELS 31
2.2 EXAMPLES OF REPRODUCING MAPPINGS 34
2.2.1 HYPERBOLIC REPRODUCING KERNELS IN THE SOBOLEV SPACE W%[A,B] 34
2.2.2 POLYNOMIAL REPRODUCING KERNELS IN THE SPACE W?[A,B] 35
2.2.3 ANALOG OF THE SPACE W^[A, B] FOR MESH FUNCTIONS 37
2.2.4 SPACE W^FO, 2IR] OF PERIODIC FUNCTIONS AND BERNOULLI FUNCTIONS 39
2.2.5 REPRODUCING KERNELS IN HUBERT SPACE OF SPHERICAL FUNCTIONS 42
2.3 SPLINE CHARACTERIZATION 44
2.3.1 GENERAL CHARACTERIZATION THEOREMS 45
2.3.2 CHARACTERIZATION FOR THE INTERPOLATION WITH COMPOSITE
INTERPOLATING OPERATOR 48
2.3.3 SMOOTHING AND MIXED SPLINES 50
3. GENERAL CONVERGENCE TECHNIQUES AND ERROR ESTIMATES FOR INTERPOLATING
SPLINES 53 3.1 GENERAL CONVERGENCE THEOREM 53
3.2 GENERAL CONVERGENCE THEOREM ON E-NETS 55
3.3 CONVERGENCE OF D M -SP MES ON SCATTERED MESHES . . .. 59
3.4 ERROR ESTIMATES FOR THE INTERPOLATING SPLINES 61
3.4.1 ERROR ESTIMATES FOR THE GENERALIZED LAGRANGIAN INTERPOLATION 61
3.4.2 SPECIAL COVERS AND ERROR ESTIMATES FOR THE SPLINES AT /I-NETS 64
3.4.3 ERROR ESTIMATES FOR Z) M -SPLINES IN LP-NORMS . . 66
4. SPLINES IN SUBSPACES 69
4.1 INTERPOLATING AND PSEUDO-INTERPOLATING SPLINES IN SUBSPACES 70
4.1.1 DEFINITIONS, ALGEBRAIC SYSTEMS 70
4.1.2 CONVERGENCE 72
4.1.3 ERROR ESTIMATES 75
4.2 SMOOTHING SPLINES IN THE SUBSPACES 78
4.2.1 DEFINITION, ALGEBRAIC SYSTEM 78
4.2.2 CONVERGENCE 78
4.2.3 ERROR ESTIMATES 80
4.2.4 ON ESTIMATION OF THE ANGLE BETWEEN SUBSPACES . 82 4.3 FINITE
ELEMENT Z? M -SPLINES AT THE SCATTERED MESHES . . 84
IMAGE 4
CONTENTS VII
4.4 DISCONTINUOUS FINITE ELEMENT M -SPLINES 87
4.4.1 DISCRETE LOCALIZATION OF DISCONTINUITIES 88
4.4.2 ACCURACY OF LOCALIZATIONS 89
4.4.3 SPECIAL FINITE ELEMENT METHOD FOR DISCONTINUOUS D M -SPLINES 91
4.4.4 NUMERICAL EXAMPLES 94
5. INTERPOLATING M -SPLINES 97
5.1 D M -SPLINES IN BOUNDED DOMAIN 98
5.1.1 INTERPOLATING Z? M -SPLINES IN ISOTROPIE AND ANISO TROPIE SOBOLEV
SPACE 98
5.1.2 UNIFORM EQUIVALENCE OF NORMS 100
5.1.3 SPECIAL COVER OF BOUNDED DOMAIN 101
5.1.4 LEMMA ON SOBOLEV FUNCTIONS WITH CONDENSED ZEROS AND CONVERGENCE
RATES FOR D M -SPLINES . . 104 5.1.5 D M -SPLINES WITH BOUNDARY
CONDITIONS 106
5.2 FINITE-ELEMENT D M -SPLINES ON SS-SPLINES 110
5.2.1 THEORETICAL GROUNDS OF APPROXIMATION WITH B SPLINES 110
5.2.2 SEMI-NORMS IN TENSOR PRODUCT OF FINITE DIMENSIONAL SPACES 111
5.2.3 POLYNOMIAL SPLINES OF THE DEFECT 1 113
5.2.4 ASSEMBLING OF INTERPOLATING MATRIX A 115
5.2.5 ASSEMBLING OF ENERGY MATRIX T 117
5.2.6 CONVERGENCE IN ANISOTROPIE SPACE 119
5.2.7 CONVERGENCE RATES IN ISOTROPIE SPACE 122
5.3 D M -SPLINES IN R N 126
5.3.1 REPRODUCING KERNEL IN D~ M L 2 126
5.3.2 INTERPOLATING SMOOTHING SPLINE 127
5.3.3 APPROXIMATION BY SPHERE INTEGRAL MEANS . . .. 128
6. SPLINES ON MANIFOLDS 135
6.1 TRACES OF Z? M -SPLINES IN UE ONTO A MANIFOLD 135
6.1.1 DEFINITIONS 136
6.1.2 SOBOLEV FUNCTIONS WITH CONDENSED ZEROS ON MANIFOLD 137
6.1.3 EXISTENCE AND UNIQUENESS 138
6.1.4 CONVERGENCE RATES 140
6.2 TRACES OF M -SPLINES IN R N ONTO A MANIFOLD 142
6.2.1 INTERPOLATING SMOOTHING SPLINES ON MANIFOLDS . 142 6.2.2 AN
ALGORITHM FOR COMPUTING THE TRACE OF D M SPLINES ON A MANIFOLD 143
6.2.3 APPROXIMATION OF SURFACES WITH KNOWN NORMALS AT THE POINTS 144
6.2.4 DISCUSSION 148
IMAGE 5
VIII
VARIATIONAL THEORY OF SPLINES
6.3 SPLINE-APPROXIMATIONS IN THIN LAYER 148
6.3.1 ANALYTICAL APPROACH 148
6.3.2 FINITE ELEMENT CASE 152
7. VECTOR SPLINES 157
7.1 CHARACTERIZATION OF VARIATIONAL VECTOR SPLINE FUNCTIONS 157 7.1.1
DIRECT SUM OF SEMI-HILBERT SPACES 157
7.1.2 ANALYTICAL REPRESENTATIONS OF VECTOR SPLINE FUNC TIONS 159
7.1.3 VECTOR SPLINES ON SUBSPACES 163
7.1.4 MERGING OF THE ANALYTIC SPLINES AND SPLINES ON SUBSPACES 166
7.1.5 SMOOTHING VECTOR SPLINE FUNCTIONS 167
7.2 RATIONAL SPLINES 168
7.2.1 OBJECT OF INTERPOLATION 168
7.2.2 INTERPOLATING RATIONAL SPLINES 169
7.2.3 CONVERGENCE OF RATIONAL ) M -SPLINES 170
7.3 APPLICATION OF VECTOR SPLINE-FUNCTIONS 172
7.3.1 CURVE APPROXIMATION BY PARAMETRIC CUBIC SPLINE 172 7.3.2 RATIONAL
SPLINES WITH THE GIVEN DERIVATIVES . . . 173 7.3.3 COLLOCATION METHOD
FOR DIFFERENTIAL EQUATIONS . 174
8. TENSOR AND BLENDING SPLINES 175
8.1 TENSOR PRODUCT OF SPACES 175
8.1.1 MAIN DEFINITIONS 175
8.1.2 A P - AND /3-NORMS 176
8.2 SOME EXTRACTS FROM GENERAL SPLINE THEORY 178
8.2.1 INTERPOLATION 178
8.2.2 SMOOTHING 179
8.3 VARIATIONAL PRINCIPLE FOR TENSOR SPLINES 180
8.3.1 SPLINE PAIRS AND SCALAR PRODUCTS 180
8.3.2 VARIATIONAL FORMULATION OF THE INTERPOLATION PROB LEM 181
8.3.3 BICUBIC SPLINES 182
8.3.4 VARIATIONAL FORMULATION OF THE SMOOTHING PROB LEM 183
8.3.5 VARIATIONAL PRINCIPLE FOR N-COMPONENT TENSOR SPLINE 184
8.4 CONVERGENCE ESTIMATES FOR TENSOR SPLINES 185
8.4.1 LIMITS OF TENSOR PRODUCTS OF OPERATORS 185
8.4.2 MAIN CONVERGENCE THEOREM 186
8.4.3 SOME APPLICATIONS OF MAIN THEOREM 187
8.5 AN ALGORITHM FOR CONSTRUCTING TENSOR SPLINES 188
8.5.1 A (T/,)-METHOD 188
8.5.2 IMPLEMENTATION OF THE TENSOR C A (U, SS)-METHOD 190
IMAGE 6
CONTENTS
IX
8.6 BLENDING SPLINES IN TENSOR PRODUCT OF SPACES 191
9. OPTIMAL APPROXIMATION OF LINEAR OPERATORS 195 9.1 GENERAL APPROACH
196
9.2 OPTIMAL APPROXIMATION OF LINEAR FUNCTIONALS 198
9.3 PROLONGATION OF MESH FUNCTIONS AND CUBATURE FORMULAS BASED ON
INDEFINITE COEFFICIENT METHOD 201
9.4 CUBATURE FORMULAS BASED ON INTERPOLATING AND SMOOTHING PROLONGATION
METHODS 203
9.4.1 LAGRANGE METHOD 203
9.4.2 INTERPOLATION AND SMOOTHING BY M -SPLINES . . 204 9.4.3
APPROXIMATION BY TRACES OF D M -SPLINES ON THE SPHERE 207
9.4.4 FINITE ELEMENT APPROXIMATION 208
9.5 EXACT INTEGRATION OF CERTAIN SPECIAL FUNCTIONS 209
9.5.1 EXACT INTEGRATION OF RADIAL FUNCTIONS X - PI 2S AND X - PI
2S IN X - P*|| ON THE UNIT SPHERE
S N -U N 3 209
9.5.2 INTEGRATION OF MONOMIALS X A ON THE UNIT SPHERE 5*_I, N 3 211
9.6 DISCUSSION 213
10. CLASSIFICATION OF SPLINE OBJECTS 215
10.1 FUNDAMENTAL OPERATIONS OVER HUBERT SPACES 216
10.1.1 CLOSED SUBSPACES AND RESTRICTION OF OPERATORS ON SUBSPACES 216
10.1.2 SPACE OF TRACES ON MANIFOLDS AND TRACE OF OPERA TOR 216
10.1.3 DIRECT SUM OF SPACES AND OPERATORS 217
10.1.4 TENSOR PRODUCT OF SPACES AND OPERATORS . . .. 217
10.1.5 CONJUGATE SPACE AND OPERATOR 218
10.2 CLASSIFICATION OF SPLINE OBJECTS AND METHODS OF THEIR MERGING 218
10.2.1 SPLINES ON SUBSPACES 219
10.2.2 SPLINES ON MANIFOLDS 219
10.2.3 VECTOR SPLINES 221
10.2.4 TENSOR SPLINES 223
10.2.5 OPTIMAL APPROXIMATION OF LINEAR FUNCTIONALS . 226
11. SN-APPROXIMATIONS AND DATA COMPRESSION 229 11.1 GENERAL
CONSIDERATION 229
11.2 OPTIMAL SUE-APPROXIMATIONS 231
11.3 EXAMPLES OF SUE-APPROXIMATIONS 237
11.3.1 TWO-DIMENSIONAL POLYNOMIAL SPLINES AND EU-APPROXIMATION 237
IMAGE 7
X
VARIATIONAL THEORY OF SPLINES
11.3.2 FOURIER EXPANSIONS AND SUE-APPROXIMATIONS . . 238 11.3.3 NUMERICAL
TESTS 240
12. ALGORITHMS FOR OPTIMAL SMOOTHING PARAMETER 243
12.1 INTRODUCTION 243
12.2 SPECTRAL DECOMPOSITION OF OPERATORS FOR SMOOTHING SPLINE PROBLEM
245
12.3 METHODS FOR CHOOSING OPTIMAL PARAMETER 248
12.3.1 NEWTON METHOD 248
12.3.2 CHEBYSHEV METHOD OF THE THIRD DEGREE 248
12.3.3 CALCULATING FORMULAS FOR DERIVATIVES OF IP(P) . . 249 12.4
DERIVATIVES OF ABSTRACT SMOOTHING SPLINE 249
12.5 DERIVATIVES OF THE SMOOTHING SPLINE ON SUBSPACE . . .. 254
12.6 DERIVATIVES OF THE SMOOTHING SPLINE BY REPRODUCING KER NELS 255
12.7 NUMERICAL FORMULAS FOR OPTIMAL SMOOTHING PARAMETER FOR DIFFERENT
ALGORITHMS 256
12.7.1 SPLINE ON SUBSPACES 256
12.7.2 SPLINES ON THE BASIS OF REPRODUCING KERNELS . . 257 12.8 UNIFORM
CONVERGENCE OF THE TAYLOR SERIES FOR SMOOTHING SPLINES 257
12.8.1 INVESTIGATION OF TAYLOR SERIES BY PARAMETER A . . 257 12.8.2
INVESTIGATION OF TAYLOR SERIES ON PARAMETER P . . 260 12.9 DISCUSSION OF
BENEFITS OF EXTRAPOLATION FOR SPLINE CONSTRUCTION ON CONVEX SET 261
APPENDICES 263
THEOREMS FROM FUNCTIONAL ANALYSIS USED IN THIS BOOK 263
A.L CONVERGENCE IN HUBERT SPACE 263
A.2 THEOREMS ON LINEAR OPERATORS 263
A.3 SOBOLEV SPACES IN DOMAIN 264
ON SOFTWARE INVESTIGATIONS IN SPLINES 265
B.L ONE-DIMENSIONAL CASE 265
B.2 MULTI-DIMENSIONAL CASE 266
BIBLIOGRAPHY 269
|
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| author_facet | Bezaev, Anatolij Ju Vasilenko, Vladimir A. |
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| building | Verbundindex |
| bvnumber | BV014180103 |
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| ctrlnum | (OCoLC)248728723 (DE-599)BVBBV014180103 |
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| id | DE-604.BV014180103 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-09T18:59:04Z |
| institution | BVB |
| isbn | 0306466422 |
| language | Undetermined |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-009718436 |
| oclc_num | 248728723 |
| open_access_boolean | |
| owner | DE-703 |
| owner_facet | DE-703 |
| physical | XVII, 280 S. |
| publishDate | 2001 |
| publishDateSearch | 2001 |
| publishDateSort | 2001 |
| publisher | Kluwer Acad./Plenum Publ. |
| record_format | marc |
| spelling | Bezaev, Anatolij Ju. Verfasser aut Variational theory of splines Anatoly Yu. Bezhaev and Vladimir A. Vasilenko New York [u.a.] Kluwer Acad./Plenum Publ. 2001 XVII, 280 S. txt rdacontent n rdamedia nc rdacarrier Spline (DE-588)4182391-6 gnd rswk-swf Variationsrechnung (DE-588)4062355-5 gnd rswk-swf Spline (DE-588)4182391-6 s Variationsrechnung (DE-588)4062355-5 s DE-604 Vasilenko, Vladimir A. Verfasser aut GBV Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009718436&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Bezaev, Anatolij Ju Vasilenko, Vladimir A. Variational theory of splines Spline (DE-588)4182391-6 gnd Variationsrechnung (DE-588)4062355-5 gnd |
| subject_GND | (DE-588)4182391-6 (DE-588)4062355-5 |
| title | Variational theory of splines |
| title_auth | Variational theory of splines |
| title_exact_search | Variational theory of splines |
| title_full | Variational theory of splines Anatoly Yu. Bezhaev and Vladimir A. Vasilenko |
| title_fullStr | Variational theory of splines Anatoly Yu. Bezhaev and Vladimir A. Vasilenko |
| title_full_unstemmed | Variational theory of splines Anatoly Yu. Bezhaev and Vladimir A. Vasilenko |
| title_short | Variational theory of splines |
| title_sort | variational theory of splines |
| topic | Spline (DE-588)4182391-6 gnd Variationsrechnung (DE-588)4062355-5 gnd |
| topic_facet | Spline Variationsrechnung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009718436&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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