Operator theory and Ill-posed problems:
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Leiden [u.a.]
VSP
2006
|
| Series: | Inverse and Ill-posed problems series
|
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Includes bibliographical references (p. [662] - 672) and index |
| Physical Description: | XVI, 680 S. |
| ISBN: | 906764448X 9789067644488 |
Staff View
MARC
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| 100 | 1 | |a Lavrentʹev, Michail Michajlovič |d 1932-2010 |e Verfasser |0 (DE-588)135733200 |4 aut | |
| 245 | 1 | 0 | |a Operator theory and Ill-posed problems |c M. M. Lavrent'ev ; L. Ya. Savel'ev |
| 264 | 1 | |a Leiden [u.a.] |b VSP |c 2006 | |
| 300 | |a XVI, 680 S. | ||
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| 338 | |b nc |2 rdacarrier | ||
| 490 | 0 | |a Inverse and Ill-posed problems series | |
| 500 | |a Includes bibliographical references (p. [662] - 672) and index | ||
| 700 | 1 | |a Savelev, Lev Jakovlevič |d 20. Jht. |e Verfasser |0 (DE-588)1089205163 |4 aut | |
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Record in the Search Index
| _version_ | 1804145315519922176 |
|---|---|
| adam_text | IMAGE 1
INVERSE AND ILL-POSED PROBLEMS SERIES
OPERATOR THEORY
AND ILL-POSED PROBLEMS
M.M. LAVRENT EV AND L.YA. SAVEL EV
NNSPIN LEIDEN * BOSTON 2006
IMAGE 2
CONTENTS
BASIC CONCEPTS L
CHAPTER 1. SET THEORY 2
1.1. SETS 2
1.1.1. ELEMENTS AND SUBSETS 2
1.1.2. THE ALGEBRA OF SETS 3
1.1.3. CARTESIAN PRODUCT 5
1.2. CORRESPONDENCES 7
1.2.1. IMAGES AND INVERSE IMAGES 7
1.2.2. FUNCTIONS 8
1.2.3. COLLECTIONS OF SETS 11
1.3. RELATIONS 14
1.3.1. REFLEXIVITY, TRANSITIVITY, AND SYMMETRY 14
1.3.2. EQUIVALENCE 15
1.3.3. ORDER 17
1.4. INDUCTION 23
1.4.1. WELL-ORDERED SETS 23
1.4.2. DISCRETE SETS 24
1.4.3. ZORN S LEMMA 26
1.5. NATURAL NUMBERS 27
1.5.1. DECIMAL NATURAL NUMBERS 27
1.5.2. THE ISOMORPHISM THEOREM 28
1.5.3. COUNTABLE SETS 29
IMAGE 3
VIII M. LAVRENT EV AND L. SAVEL EV. OPERATOR THEORY AND ILL-POSED ...
CHAPTER 2. ALGEBRA 30
2.1. ABSTRACT ALGEBRA 30
2.1.1. SEMIGROUPS 30
2.1.2. GROUPS 37
2.1.3. RINGS AND FIELDS 55
2.1.4. LATTICES 64
2.1.5. NUMBERS 69
2.2. LINEAR ALGEBRA 75
2.2.1. VECTOR SPACES 76
2.2.2. LINEAR OPERATORS 89
2.2.3. LINEAR FUNCTIONALS 102
2.2.4. SCALAR PRODUCTS 114
2.2.5. NORMED SPACES 121
2.2.6. EUCLIDEAN SPACES 130
2.3. MULTILINEAR ALGEBRA 141
2.3.1. TENSOR PRODUCT 141
2.3.2. EXTERIOR PRODUCT 146
CHAPTER 3. CALCULUS 150
3.1. LIMIT 150
3.1.1. TOPOLOGICAL SPACES 150
3.1.2. DIRECTED SETS 170
3.1.3. CONVERGENCE 177
3.2. DIFFERENTIAL 197
3.2.1. THE DEFINITION OF THE DIFFERENTIAL 197
3.2.2. DIFFERENTIATION RULES 206
3.2.3. LAGRANGE S THEOREM 210
3.2.4. TERMWISE DIFFERENTIATION 215
3.2.5. TOTAL DIFFERENTIALS 217
3.2.6. SOLUTION OF FUNCTIONAL EQUATIONS 226
3.2.7. TAYLOR S FORMULA 241
3.2.8. LOCAL MINIMA 250
3.2.9. SMOOTH CURVES 258
3.2.10. A SIMPLEST VARIATIONAL PROBLEM 261
IMAGE 4
CONTENTS IX
3.3. INTEGRAL 263
3.3.1. MEASURES 264
3.3.2. CLASSICAL DEFINITION OF THE INTEGRAL 272
3.3.3. LIMIT THEOREMS 289
3.3.4. MEASURABLE FUNCTIONS 295
3.3.5. THE FUBINI AND TONELLI THEOREMS 298
3.3.6. INDEFINITE INTEGRALS 316
3.4. ANALYSIS ON MANIFOLDS 321
3.4.1. MANIFOLDS 321
3.4.2. THE RANK THEOREM 326
3.4.3. SARD S THEOREM 327
3.4.4. DIFFERENTIAL FORMS 328
3.4.5. THE POINCARE THEOREM 331
3.4.6. CHANGE OF VARIABLES 334
3.4.7. INTEGRAL OVER A MANIFOLD 335
3.4.8. THE STOKES FORMULA 342
3.4.9. MAP DEGREE 348
3.4.10. APPLICATIONS 351
OPERATORS 353
CHAPTER 4. LINEAR OPERATORS 354
4.1. HILBERT SPACES 354
4.1.1. ORTHOGONAL PROJECTION 354
4.1.2. CONTINUOUS LINEAR FUNCTIONALS 356
4.1.3. THE SPACES C 2 = JC 2 (U,N) 359
4.2. FOURIER SERIES 363
4.2.1. FOURIER COEFFICIENTS 363
4.2.2. ISOMORPHISM OF HILBERT SPACES 368
4.3. FUNCTION SPACES 369
4.3.1. METRIC SPACES 369
4.3.2. SMOOTH FUNCTIONS 371
4.3.3. LEBESGUE SPACES 376
4.3.4. DISTRIBUTIONS 380
4.3.5. SOBOLEV SPACES 388
IMAGE 5
M. LAVRENT EV AND L. SAVEL EV. OPERATOR THEORY AND ILL-POSED ...
4.4. FOURIER TRANSFORM 391
4.4.1. TRANSFORMS OF RAPIDLY DECREASING FUNCTIONS 391 4.4.2. TRANSFORMS
OF SLOWLY INCREASING DISTRIBUTIONS 393 4.4.3. THE FOURIER-PLANCHEREL
TRANSFORM 395
4.4.4. THE FOURIER-STIELTJES TRANSFORM 395
4.4.5. THE RADON TRANSFORM 396
4.5. BOUNDED LINEAR OPERATORS 397
4.5.1. EXTENSIONS OF FUNCTIONALS 397
4.5.2. UNIFORM BOUNDEDNESS OF OPERATORS 399
4.5.3. INVERSION OF OPERATORS 401
4.5.4. CLOSEDNESS OF THE GRAPH OF AN OPERATOR 403
4.5.5. WEAK COMPACTNESS 405
4.6. COMPACT LINEAR OPERATORS 408
4.6.1. EXAMPLES OF COMPACT OPERATORS 408
4.6.2. PROPERTIES OF COMPACT OPERATORS 410
4.6.3. ADJOINT OPERATORS 411
4.6.4. FREDHOLM OPERATORS 414
4.6.5. FREDHOLM THEOREMS 417
4.7. SELF-ADJOINT OPERATORS 419
4.7.1. BANACH ADJOINT OPERATORS 419
4.7.2. HILBERT ADJOINT OPERATORS 420
4.7.3. HERMITIAN AND NORMAL OPERATORS 422
4.7.4. UNITARY OPERATORS 423
4.7.5. POSITIVE OPERATORS 424
4.8. SPECTRA OF OPERATORS 426
4.8.1. CLASSIFICATION OF SPECTRA 426
4.8.2. THE SPECTRUM OF A CLOSED OPERATOR 431
4.8.3. THE SPECTRUM OF A BOUNDED OPERATOR 433
4.8.4. THE SPECTRUM OF A COMPACT OPERATOR 435
4.8.5. THE SPECTRUM OF A SELF-ADJOINT OPERATOR 435
4.9. SPECTRAL THEOREM 444
4.9.1. PROJECTION MEASURES 445
4.9.2. INTEGRALS OF BOUNDED FUNCTIONS 451
4.9.3. INTEGRALS OF UNBOUNDED FUNCTIONS 459
4.9.4. SPECTRAL THEOREM 462
4.9.5. OPERATOR FUNCTIONS 466
IMAGE 6
CONTENTS XI
4.10. OPERATOR EXPONENTIAL 468
4.10.1. PROBLEM FORMULATION 468
4.10.2. SEMIGROUPS OF OPERATORS 470
4.10.3. THE LAPLACE TRANSFORM 475
4.10.4. STONE S THEOREM 476
4.10.5. EVOLUTION EQUATIONS 478
CHAPTER 5. NONLINEAR OPERATORS 483
5.1. FIXED POINTS 483
5.1.1. THE BROUWER THEOREM 483
5.1.2. THE TIKHONOV THEOREM AND THE SCHAUDER THEOREM . . . 487 5.2.
SADDLE POINTS 490
5.2.1. KAKUTANI S THEOREM 490
5.2.2. VON NEUMANN THEOREM 493
5.3. MONOTONIC OPERATORS 497
5.3.1. DEFINITION AND PROPERTIES 497
5.3.2. EQUATIONS WITH MONOTONIC OPERATORS . 499
5.4. NONLINEAR CONTRACTIONS 501
5.4.1. CONTRACTING SEMIGROUPS OF OPERATORS 501
5.4.2. APPROXIMATION 503
5.5. DEGREE THEORY 504
5.5.1. FINITE-DIMENSIONAL SPACES 504
5.5.2. THE LERAY-SCHAUDER DEGREE 508
ILL-POSED PROBLEMS 511
CHAPTER 6. CLASSIC PROBLEMS 512
6.1. MATHEMATICAL DESCRIPTION OF THE LAWS OF PHYSICS 512 6.2. EQUATIONS
OF THE FIRST ORDER 518
6.3. CLASSIFICATION OF DIFFERENTIAL EQUATIONS OF THE SECOND ORDER . . .
519 6.4. ELLIPTIC EQUATIONS 521
6.5. HYPERBOLIC AND PARABOLIC EQUATIONS 527
6.6. THE NOTION OF WELL-POSEDNESS 529
IMAGE 7
XII M. LAVRENT EV AND L. SAVEL EV. OPERATOR THEORY AND ILL-POSED ...
CHAPTER 7. ILL-POSED PROBLEMS 531
7.1. ILL-POSED CAUCHY PROBLEMS 531
7.2. ANALYTIC CONTINUATION AND INTERIOR PROBLEMS 534
7.3. WEAKLY AND STRONGLY ILL-POSED PROBLEMS. PROBLEMS OF DIFFEREN-
TIATION 536
7.4. REDUCING ILL-POSED PROBLEMS TO INTEGRAL EQUATIONS 537
CHAPTER 8. PHYSICAL PROBLEMS LEADING TO ILL-POSED PROBLEMS 541 8.1.
INTERPRETATION OF MEASUREMENT DATA FROM PHYSICAL DEVICES . . . 541 8.2.
INTERPRETATION OF GRAVIMETRIC DATA 543
8.3. PROBLEMS FOR THE DIFFUSION EQUATION 546
8.4. DETERMINING PHYSICAL FIELDS FROM THE MEASUREMENTS DATA . . . 547
8.5. TOMOGRAPHY 548
CHAPTER 9. OPERATOR AND INTEGRAL EQUATIONS 552
9.1. DEFINITIONS OF WELL-POSEDNESS 552
9.2. REGULARIZATION 555
9.3. LINEAR OPERATOR EQUATIONS 559
9.4. INTEGRAL EQUATIONS WITH WEAK SINGULARITIES 564
9.5. SCALAR VOLTERRA EQUATIONS 565
9.6. VOLTERRA OPERATOR EQUATIONS 568
CHAPTER 10. EVOLUTION EQUATIONS 571
10.1. CAUCHY PROBLEM AND SEMIGROUPS OF OPERATORS 571 10.2. EQUATIONS IN
A HILBERT SPACE 573
10.3. EQUATIONS WITH VARIABLE OPERATOR 577
10.4. EQUATIONS OF THE SECOND ORDER 578
10.5. WELL-POSED AND ILL-POSED CAUCHY PROBLEMS 580
10.6. EQUATIONS WITH INTEGRO-DIFFERENTIAL OPERATORS 581
CHAPTER 11. PROBLEMS OF INTEGRAL GEOMETRY 584
11.1. STATEMENT OF PROBLEMS OF INTEGRAL GEOMETRY 584
11.2. THE RADON PROBLEM 584
11.3. RECONSTRUCTING A FUNCTION FROM SPHERICAL MEANS 588 11.4. PLANAR
PROBLEM OF THE GENERAL FORM 594
11.5. SPATIAL PROBLEMS OF THE GENERAL FORM 602
IMAGE 8
CONTENTS XIII
11.6. PROBLEMS OF THE VOLTERRA TYPE FOR MANIFOLDS INVARIANT WITH RESPECT
TO THE TRANSLATION GROUP 614
11.7. PLANAR PROBLEMS OF INTEGRAL GEOMETRY WITH A PERTURBATION . . 618
CHAPTER 12. INVERSE PROBLEMS 626
12.1. STATEMENT OF INVERSE PROBLEMS 626
12.2. INVERSE DYNAMIC PROBLEM. A LINEARIZATION METHOD 628 12.3. A
GENERAL METHOD FOR STUDYING INVERSE PROBLEMS FOR HYPER- BOLIC EQUATIONS
637
12.4. THE CONNECTION BETWEEN INVERSE PROBLEMS FOR HYPERBOLIC, ELLIPTIC,
AND PARABOLIC EQUATIONS 644
12.5. PROBLEMS OF DETERMINING A RIEMANNIAN METRIC 651
CHAPTER 13. SEVERAL AREAS OF THE THEORY OF
ILL-POSED PROBLEMS, INVERSE PROBLEMS, AND APPLICATIONS 659
BIBLIOGRAPHY 662
INDEX 673
|
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| author | Lavrentʹev, Michail Michajlovič 1932-2010 Savelev, Lev Jakovlevič 20. Jht |
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| isbn | 906764448X 9789067644488 |
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| spelling | Lavrentʹev, Michail Michajlovič 1932-2010 Verfasser (DE-588)135733200 aut Operator theory and Ill-posed problems M. M. Lavrent'ev ; L. Ya. Savel'ev Leiden [u.a.] VSP 2006 XVI, 680 S. txt rdacontent n rdamedia nc rdacarrier Inverse and Ill-posed problems series Includes bibliographical references (p. [662] - 672) and index Savelev, Lev Jakovlevič 20. Jht. Verfasser (DE-588)1089205163 aut GBV Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=022247049&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Lavrentʹev, Michail Michajlovič 1932-2010 Savelev, Lev Jakovlevič 20. Jht Operator theory and Ill-posed problems |
| title | Operator theory and Ill-posed problems |
| title_auth | Operator theory and Ill-posed problems |
| title_exact_search | Operator theory and Ill-posed problems |
| title_full | Operator theory and Ill-posed problems M. M. Lavrent'ev ; L. Ya. Savel'ev |
| title_fullStr | Operator theory and Ill-posed problems M. M. Lavrent'ev ; L. Ya. Savel'ev |
| title_full_unstemmed | Operator theory and Ill-posed problems M. M. Lavrent'ev ; L. Ya. Savel'ev |
| title_short | Operator theory and Ill-posed problems |
| title_sort | operator theory and ill posed problems |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=022247049&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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