Theoretical numerical analysis: a functional analysis framework
Gespeichert in:
| Beteiligte Personen: | , |
|---|---|
| Format: | Buch |
| Sprache: | Englisch |
| Veröffentlicht: |
Berlin [u.a.]
Springer
2005
|
| Ausgabe: | 2. ed. |
| Schriftenreihe: | Texts in applied mathematics
39 |
| Schlagwörter: | |
| Links: | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=013181433&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| Umfang: | XVIII, 576 S. graph. Darst. |
| ISBN: | 0387258876 9780387258874 |
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Datensatz im Suchindex
| _version_ | 1819364045443039232 |
|---|---|
| adam_text | KENDALI ATKINSON WEIMIN HAN THEORETICAL NUMERICAL ANALYSIS A FUNCTIONAL
ANALYSIS FRAMEWORK SECOND EDITION WITH 36 ILLUSTRATIONS SPRINGER
CONTENTS SERIES PREFACE VII PREFACE TO THE SECOND EDITION IX PREFACE TO
THE FIRST EDITION XI 1 LINEAR SPACES 1 1.1 LINEAR SPACES 1 1.2 NORMED
SPACES 7 1.2.1 CONVERGENCE 10 1.2.2 BANACH SPACES 13 1.2.3 COMPLETION OF
NORMED SPACES 14 1.3 INNER PRODUCT SPACES 21 1.3.1 HUBERT SPACES 26
1.3.2 ORTHOGONALITY 28 1.4 SPACES OF CONTINUOUSLY DIFFERENTIABLE
FUNCTIONS 38 1.4.1 HOLDER SPACES 41 1.5 L P SPACES 43 1.6 COMPACT SETS
47 2 LINEAR OPERATORS ON NORMED SPACES 51 2.1 OPERATORS 52 2.2
CONTINUOUS LINEAR OPERATORS 55 2.2.1 C{V,W) AS A BANACH SPACE 59 XIV
CONTENTS 2.3 THE GEOMETRIC SERIES THEOREM AND ITS VARIANTS 60 2.3.1 A
GENERALIZATION 64 2.3.2 A PERTURBATION RESULT 66 2.4 SOME MORE RESULTS
ON LINEAR OPERATORS 71 2.4.1 AN EXTENSION THEOREM 72 2.4.2 OPEN MAPPING
THEOREM 73 2.4.3 PRINCIPLE OF UNIFORM BOUNDEDNESS 75 2.4.4 CONVERGENCE
OF NUMERICAL QUADRATURES 76 2.5 LINEAR FUNCTIONALS 79 2.5.1 AN EXTENSION
THEOREM FOR LINEAR FUNCTIONALS 80 2.5.2 THE RIESZ REPRESENTATION THEOREM
82 2.6 ADJOINT OPERATORS 85 2.7 TYPES OF CONVERGENCE 90 2.8 COMPACT
LINEAR OPERATORS 93 2.8.1 COMPACT INTEGRAL OPERATORS ON C{D) 94 2.8.2
PROPERTIES OF COMPACT OPERATORS 95 2.8.3 INTEGRAL OPERATORS ON L 2 (A,
B) 97 2.8.4 THE FREDHOLM ALTERNATIVE THEOREM 99 2.8.5 ADDITIONAL RESULTS
ON FREDHOLM INTEGRAL EQUATIONS . 103 2.9 THE RESOLVENT OPERATOR 107
2.9.1 R(X) AS A HOLOMORPHIC FUNCTION 108 3 APPROXIMATION THEORY 113 3.1
APPROXIMATION OF CONTINUOUS FUNCTIONS BY POLYNOMIALS ... 114 3.2
INTERPOLATION THEORY 115 3.2.1 LAGRANGE POLYNOMIAL INTERPOLATION 117
3.2.2 HERMITE POLYNOMIAL INTERPOLATION 121 3.2.3 PIECEWISE POLYNOMIAL
INTERPOLATION 121 3.2.4 TRIGONOMETRIE INTERPOLATION 124 3.3 BEST
APPROXIMATION 129 3.3.1 CONVEXITY, LOWER SEMICONTINUITY 129 3.3.2 SOME
ABSTRACT EXISTENCE RESULTS 131 3.3.3 EXISTENCE OF BEST APPROXIMATION 134
3.3.4 UNIQUENESS OF BEST APPROXIMATION 136 3.4 BEST APPROXIMATIONS IN
INNER PRODUCT SPACES, PROJECTION ON CLOSED CONVEX SETS 139 3.5
ORTHOGONAL POLYNOMIALS 146 3.6 PROJECTION OPERATORS 150 3.7 UNIFORM
ERROR BOUNDS 154 3.7.1 UNIFORM ERROR BOUNDS FOR L 2 -APPROXIMATIONS . .
. . 156 3.7.2 INTERPOLATORY PROJEETIONS AND THEIR CONVERGENCE . . . 158
4 FOURIER ANALYSIS AND WAVELETS 161 4.1 FOURIER SERIES 161 4.2 FOURIER
TRANSFORM 175 CONTENTS XV 4.3 DISCRETE FOURIER TRANSFORM 180 4.4 HAAR
WAVELETS 185 4.5 MULTIRESOLUTION ANALYSIS 193 5 NONLINEAR EQUATIONS AND
THEIR SOLUTION BY ITERATION 201 5.1 THE BANACH FIXED-POINT THEOREM 202
5.2 APPLICATIONS TO ITERATIVE METHODS 206 5.2.1 NONLINEAR EQUATIONS 207
5.2.2 LINEAR SYSTEMS 208 5.2.3 LINEAR AND NONLINEAR INTEGRAL EQUATIONS
211 5.2.4 ORDINARY DIFFERENTIAL EQUATIONS IN BANACH SPACES . . 215 5.3
DIFFERENTIAL CALCULUS FOR NONLINEAR OPERATORS 219 5.3.1 FRECHET AND
GAETEAUX DERIVATIVES 219 5.3.2 MEAN VALUE THEOREMS 223 5.3.3 PARTIAL
DERIVATIVES 224 5.3.4 THE GAETEAUX DERIVATIVE AND CONVEX MINIMIZATION . .
226 5.4 NEWTON S METHOD 230 5.4.1 NEWTON S METHOD IN A BANACH SPACE 230
5.4.2 APPLICATIONS 233 5.5 COMPLETELY CONTINUOUS VECTOR FIELDS 236 5.5.1
THE ROTATION OF A COMPLETELY CONTINUOUS VECTOR FIELD 238 5.6 CONJUGATE
GRADIENT METHOD FOR OPERATOR EQUATIONS 239 6 FINITE DIFFERENCE METHOD
249 6.1 FINITE DIFFERENCE APPROXIMATIONS 249 6.2 LAX EQUIVALENCE THEOREM
256 6.3 MORE ON CONVERGENCE 265 7 SOBOLEV SPACES 273 7.1 WEAK
DERIVATIVES 273 7.2 SOBOLEV SPACES 279 7.2.1 SOBOLEV SPACES OF INTEGER
ORDER 280 7.2.2 SOBOLEV SPACES OF REAL ORDER 286 7.2.3 SOBOLEV SPACES
OVER BOUNDARIES 288 7.3 PROPERTIES 289 7.3.1 APPROXIMATION BY SMOOTH
FUNCTIONS 290 7.3.2 EXTENSIONS 291 7.3.3 SOBOLEV EMBEDDING THEOREMS 291
7.3.4 TRACES 293 7.3.5 EQUIVALENT NORMS 294 7.3.6 A SOBOLEV QUOTIENT
SPACE 298 7.4 CHARACTERIZATION OF SOBOLEV SPACES VIA THE FOURIER
TRANSFORM 303 7.5 PERIODIC SOBOLEV SPACES 307 7.5.1 THE DUAL SPACE . . .
309 7.5.2 EMBEDDING RESULTS 310 XVI CONTENTS 7.5.3 APPROXIMATION RESULTS
311 7.5.4 AN ILLUSTRATIVE EXAMPLE OF AN OPERATOR 312 7.5.5 SPHERICAL
POLYNOMIALS AND SPHERICAL HARMONICS . . . 313 7.6 INTEGRATION BY PARTS
FORMULAS 319 8 VARIATIONAL FORMULATIONS OF ELLIPTIC BOUNDARY VALUE PROB-
LEMS 323 8.1 A MODEL BOUNDARY VALUE PROBLEM 324 8.2 SOME GENERAL RESULTS
ON EXISTENCE AND UNIQUENESS 326 8.3 THE LAX-MILGRAM LEMMA 330 8.4 WEAK
FORMULATIONS OF LINEAR ELLIPTIC BOUNDARY VALUE PROBLEMS 334 8.4.1
PROBLEMS WITH HOMOGENEOUS DIRICHLET BOUNDARY CON- DITIONS 334 8.4.2
PROBLEMS WITH NON-HOMOGENEOUS DIRICHLET BOUNDARY CONDITIONS 335 8.4.3
PROBLEMS WITH NEUMANN BOUNDARY CONDITIONS . . . . 337 8.4.4 PROBLEMS
WITH MIXED BOUNDARY CONDITIONS 339 8.4.5 A GENERAL LINEAR SECOND-ORDER
ELLIPTIC BOUNDARY VALUE PROBLEM 340 8.5 A BOUNDARY VALUE PROBLEM OF
LINEARIZED ELASTICITY 343 8.6 MIXED AND DUAL FORMULATIONS 348 8.7
GENERALIZED LAX-MILGRAM LEMMA 353 8.8 A NONLINEAR PROBLEM 355 9 THE
GALERKIN METHOD AND ITS VARIANTS 361 9.1 THE GALERKIN METHOD 361 9.2 THE
PETROV-GALERKIN METHOD 367 9.3 GENERALIZED GALERKIN METHOD 370 9.4
CONJUGATE GRADIENT METHOD: VARIATIONAL FORMULATION . . . . 372 10 FINITE
ELEMENT ANALYSIS 377 10.1 ONE-DIMENSIONAL EXAMPLES 379 10.1.1 LINEAR
ELEMENTS FOR A SECOND-ORDER PROBLEM 379 10.1.2 HIGH ORDER ELEMENTS AND
THE CONDENSATION TECHNIQUE 382 10.1.3 REFERENCE ELEMENT TECHNIQUE,
NON-CONFORMING METHOD 384 10.2 BASICS OF THE FINITE ELEMENT METHOD 387
10.2.1 TRIANGULATION 387 10.2.2 POLYNOMIAL SPACES ON THE REFERENCE
ELEMENTS 389 10.2.3 AFFINE-EQUIVALENT FINITE ELEMENTS 391 10.2.4 FINITE
ELEMENT SPACES 392 10.2.5 INTERPOLATION 395 10.3 ERROR ESTIMATES OF
FINITE ELEMENT INTERPOLATIONS 396 10.3.1 INTERPOLATION ERROR ESTIMATES
ON THE REFERENCE ELEMENT 397 10.3.2 LOCAL INTERPOLATION ERROR ESTIMATES
398 10.3.3 GLOBAL INTERPOLATION ERROR ESTIMATES 401 CONTENTS XVII 10.4
CONVERGENCE AND ERROR ESTIMATES 404 11 ELLIPTIC VARIATIONAL INEQUALITIES
AND THEIR NUMERICAL AP- PROXIMATIONS 413 11.1 INTRODUCTORY EXAMPLES 413
11.2 ELLIPTIC VARIATIONAL INEQUALITIES OF THE FIRST KIND 420 11.3
APPROXIMATION OF EVIS OF THE FIRST KIND . 425 11.4 ELLIPTIC VARIATIONAL
INEQUALITIES OF THE SECOND KIND 428 11.5 APPROXIMATION OF EVIS OF THE
SECOND KIND 434 11.5.1 REGULARIZATION TECHNIQUE 436 11.5.2 METHOD OF
LAGRANGIAN MULTIPLIERS 438 11.5.3 METHOD OF NUMERICAL INTEGRATION 440 12
NUMERICAL SOLUTION OF FREDHOLM INTEGRAL EQUATIONS OF THE SECOND KIND 447
12.1 PROJECTION METHODS: GENERAL THEORY 448 12.1.1 COLLOCATION METHODS
448 12.1.2 GALERKIN METHODS 450 12.1.3 A GENERAL THEORETICAL FRAMEWORK
451 12.2 EXAMPLES 456 12.2.1 PIECEWISE LINEAR COLLOCATION 457 12.2.2
TRIGONOMETRIC POLYNOMIAL COLLOCATION 459 12.2.3 A PIECEWISE LINEAR
GALERKIN METHOD 461 12.2.4 A GALERKIN METHOD WITH TRIGONOMETRIC
POLYNOMIALS . 463 12.3 ITERATED PROJECTION METHODS 468 12.3.1 THE
ITERATED GALERKIN METHOD 470 12.3.2 THE ITERATED COLLOCATION SOLUTION
472 12.4 THE NYSTROEM METHOD 478 12.4.1 THE NYSTROEM METHOD FOR CONTINUOUS
KERNEL FUNCTIONS 478 12.4.2 PROPERTIES AND ERROR ANALYSIS OF THE NYSTROEM
METHOD 481 12.4.3 COLLECTIVELY COMPACT OPERATOR APPROXIMATIONS . . . .
489 12.5 PRODUCT INTEGRATION 492 12.5.1 ERROR ANALYSIS 494 12.5.2
GENERALIZATIONS TO OTHER KERNEL FUNCTIONS 496 12.5.3 IMPROVED ERROR
RESULTS FOR SPECIAL KERNEIS 498 12.5.4 PRODUCT INTEGRATION WITH GRADED
MESHES 498 12.5.5 THE RELATIONSHIP OF PRODUCT INTEGRATION AND COLLOCA-
TION METHODS 503 12.6 ITERATION METHODS 504 12.6.1 A TWO-GRID ITERATION
METHOD FOR THE NYSTROEM METHOD 505 12.6.2 CONVERGENCE ANALYSIS 508 12.6.3
THE ITERATION METHOD FOR THE LINEAR SYSTEM 511 12.6.4 AN OPERATIONS
COUNT 513 12.7 PROJECTION METHODS FOR NONLINEAR EQUATIONS 515 12.7.1
LINEARIZATION 515 XVIII CONTENTS 12.7.2 A HOMOTOPY ARGUMENT 518 12.7.3
THE APPROXIMATING FINITE-DIMENSIONAL PROBLEM . . . 520 13 BOUNDARY
INTEGRAL EQUATIONS 523 13.1 BOUNDARY INTEGRAL EQUATIONS 524 13.1.1
GREEN S IDENTITIES AND REPRESENTATION FORMULA . . . . 525 13.1.2 THE
KELVIN TRANSFORMATION AND EXTERIOR PROBLEMS . 527 13.1.3 BOUNDARY
INTEGRAL EQUATIONS OF DIRECT TYPE 531 13.2 BOUNDARY INTEGRAL EQUATIONS
OF THE SECOND KIND 537 13.2.1 EVALUATION OF THE DOUBLE LAYER POTENTIAL
540 13.2.2 THE EXTERIOR NEUMANN PROBLEM 543 13.3 A BOUNDARY INTEGRAL
EQUATION OF THE FIRST KIND 549 13.3.1 A NUMERICAL METHOD 551 REFERENCES
555 INDEX 569
|
| any_adam_object | 1 |
| author | Atkinson, Kendall E. 1940- Han, Weimin 1963- |
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| discipline | Mathematik |
| edition | 2. ed. |
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| id | DE-604.BV019856853 |
| illustrated | Illustrated |
| indexdate | 2024-12-20T12:07:22Z |
| institution | BVB |
| isbn | 0387258876 9780387258874 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-013181433 |
| oclc_num | 61659216 |
| open_access_boolean | |
| owner | DE-20 DE-824 DE-355 DE-BY-UBR DE-29T DE-11 DE-83 |
| owner_facet | DE-20 DE-824 DE-355 DE-BY-UBR DE-29T DE-11 DE-83 |
| physical | XVIII, 576 S. graph. Darst. |
| publishDate | 2005 |
| publishDateSearch | 2005 |
| publishDateSort | 2005 |
| publisher | Springer |
| record_format | marc |
| series | Texts in applied mathematics |
| series2 | Texts in applied mathematics |
| spellingShingle | Atkinson, Kendall E. 1940- Han, Weimin 1963- Theoretical numerical analysis a functional analysis framework Texts in applied mathematics Analyse fonctionnelle Análise funcional larpcal Análise numérica larpcal Functional analysis Funktionalanalysis (DE-588)4018916-8 gnd Numerische Mathematik (DE-588)4042805-9 gnd |
| subject_GND | (DE-588)4018916-8 (DE-588)4042805-9 |
| title | Theoretical numerical analysis a functional analysis framework |
| title_auth | Theoretical numerical analysis a functional analysis framework |
| title_exact_search | Theoretical numerical analysis a functional analysis framework |
| title_full | Theoretical numerical analysis a functional analysis framework Kendall Atkinson ; Weimin Han |
| title_fullStr | Theoretical numerical analysis a functional analysis framework Kendall Atkinson ; Weimin Han |
| title_full_unstemmed | Theoretical numerical analysis a functional analysis framework Kendall Atkinson ; Weimin Han |
| title_short | Theoretical numerical analysis |
| title_sort | theoretical numerical analysis a functional analysis framework |
| title_sub | a functional analysis framework |
| topic | Analyse fonctionnelle Análise funcional larpcal Análise numérica larpcal Functional analysis Funktionalanalysis (DE-588)4018916-8 gnd Numerische Mathematik (DE-588)4042805-9 gnd |
| topic_facet | Analyse fonctionnelle Análise funcional Análise numérica Functional analysis Funktionalanalysis Numerische Mathematik |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=013181433&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV002476038 |
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