The commutant lifting approach to interpolation problems:
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Basel u.a.
Birkhäuser
1990
|
| Series: | Operator theory
44 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XXIII, 632 S. |
| ISBN: | 3764324619 0817624619 |
Staff View
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| 245 | 1 | 0 | |a The commutant lifting approach to interpolation problems |c Ciprian Foias ; Arthur E. Frazho |
| 264 | 1 | |a Basel u.a. |b Birkhäuser |c 1990 | |
| 300 | |a XXIII, 632 S. | ||
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Record in the Search Index
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|---|---|
| adam_text | TABLE OF CONTENTS
Page
CHAPTER I. ANALYSIS OF THE CARATHEODORY
INTERPOLATION PROBLEM 1
1. THE CARATHEODORY INTERPOLATION PROBLEM AND SCHUR
NUMBERS 1
2. BLASCHKE PRODUCTS, UNIQUE EXTENSIONS AND THE
JURY TEST 7
3. THE SCHUR REPRESENTATION 11
4. COMPUTING SCHUR NUMBERS 16
5. COMPUTING THE SCHUR NUMBERS FROM I AnA^ 20
6. CONTRACTIONS AND THE CARATHEODORY FEJER
THEOREMS 24
7. THE SCHUR COHN TEST 30
8. NOTES AND COMMENTS 32
CHAPTER H. ANALYSIS OF THE CARATHEODORY INTERPOLATION
PROBLEM FOR POSITIVE REAL FUNCTIONS 33
1. POSITIVE REAL FUNCTIONS 33
2. THE LEVINSON ALGORITHM 37
3. UNIQUENESS AND POSITIVE EXTENSIONS 41
4. COMPUTING SCHUR NUMBERS FROM THE LEVINSON
ALGORITHM 43
5. MAXIMAL ENTROPY 47
6. NOTES AND COMMENTS 49
xvi TABLE OF CONTENTS
Page
CHAPTER m. SCHUR NUMBERS, GEOPHYSICS AND INVERSE
SCATTERING PROBLEMS 53
1. THE LAYERED MEDIUM MODEL 53
2. SCHUR NUMBERS AND REFLECTION COEFFICIENTS 55
3. COMPUTING WITH THE LAYERED MEDIUM MODEL 58
4. THE SCHUR REPRESENTATION AND THE LAYERED
MEDIUM MODEL 61
5. THE LEVINSON ALGORITHM AND MARINE SEISMOLOGY 65
6. SCATTERING AND ROTATION MATRICES 68
7. NOTES AND COMMENTS 70
CHAPTER IV. CONTRACTIVE EXPANSIONS ON EUCLIDIAN AND
HILBERT SPACE 71
1. CONTRACTIVE 2 by 1 COLUMN MATRICES 71
2. CONTRACTIVE 2 by 2 LOWER TRIANGULAR MATRICES 76
3. CONTRACTIVE 2 by 2 MATRICES 77
4. COMPUTING ALL CONTRACTIVE 2 by 2 MATRICES 80
5. CONTRACTIVE LEVINSON SYSTEMS 84
6. EXPANDING MATRIX CONTRACTIONS 86
7. THE CARATHEODORY INTERPOLATION PROBLEM
REVISITED 91
8. NOTES AND COMMENTS 97
CHAPTER V. CONTRACTIVE ONE STEP INTERTWINING
LIFTINGS 99
1. CHARACTERIZING ALL CONTRACTIVE ONE STEP
INTERTWINING LIFTINGS 99
2. REGULAR FACTORIZATION 106
TABLE OF CONTENTS xvi i
Page
3. APPLICATIONS OF CONTRACTIVE ONE STEP LIFTINGS
TO CARATHEODORY INTERPOLATION 110
4. A 2 by 2 MATRIX CHARACTERIZATION OF ALL CONTRACTIVE
ONE STEP INTERTWINING LIFTINGS 112
5. THE SCHUR COHN TEST REVISITED 117
6. NOTES AND COMMENTS 120
CHAPTER VL ISOMETRIC AND UNITARY DILATIONS 123
1. THE WOLD DECOMPOSITION 123
2. BILATERAL SHIFTS 127
3. MINIMAL ISOMETRIC DILATIONS 131
4. THE GEOMETRY OF MINIMAL ISOMETRIC DILATIONS 134
5. MINIMAL UNITARY DILATIONS 138
6. COMPLETELY NON UNITARY CONTRACTIONS 141
7. NON MINIMAL ISOMETRIC DILATIONS 146
8. NOTES AND COMMENTS 151
CHAPTER VH. THE COMMUTANT LIFTING THEOREM 153
1. THE COMMUTANT LIFTING THEOREM 153
2. STRONG LIMITS OF CONTRACTIVE LIFTINGS 158
3. THE SECOND PROOF OF THE COMMUTANT LIFTING
THEOREM 162
4. CYCLIC SPACES AND UNIQUE CONTRCTIVE INTERTWINING
LIFTINGS 164
5. REGULAR FACTORIZATIONS AND UNIQUE CONTRACTIVE
INTERTWINING LIFTINGS 170
6. COMMUTING ISOMETRIC LIFTINGS AND THE THIRD PROOF
OF THE COMMUTANT LIFTING THEOREM 173
xvi i i TABLE OF CONTENTS
Page
7. THE COUPLING OF T, A, AND T 177
8. THE FOURTH PROOF OF THE COMMUTANT LIFTING
THEOREM 181
9. ANOTHER METHOD OF CONSTRUCTING ALL CONTRACTIVE
ONE STEP COUPLINGS 187
10. NOTES AND COMMENTS 189
CHAPTER VIII. GEOMETRIC APPLICATIONS OF THE
COMMUTANT LIFTING THEOREM 191
1. OPERATOR VALUED CARATHEODORY INTERPOLATION 191
2. OPERATOR VALUED NEVANLINNA PICK INTERPOLATION 197
3. THE CARSWELL SCHUBERT RESULT 202
4. TOEPLITZ AND HANKEL OPERATORS 207
5. INVERTIBLE TOEPLITZ OPERATORS 211
6. CORONA THEOREMS FOR ANALYTIC TOEPLITZ
OPERATORS 217
7. GENERALIZED HANKEL MATRICES AND CONTRACTIVE ONE
STEP INTERTWINING LIFTING 223
8. NOTES AND COMMENTS 231
CHAPTER IX. H~ OPTIMIZATION AND FUNCTIONAL MODELS 233
1. MULTIPLICATION OPERATORS 233
2. THE BEURLING LAX HALMOS THEOREM 239
3. HANKEL OPERATORS AND THE NEHARI PROBLEM 242
4. H~ OPTIMIZATION 247
5. THE BASIC FUNCTIONAL MODEL 252
6. THE CHARACTERISTIC FUNCTION 258
7. THE OPERATORS A(F) 262
TABLE OF CONTENTS xix
Page
8. THE DOUGLAS SHAPIRO SHIELDS FACTORIZATION 267
9. CHARACTERIZING 7(S(0),S(01)) 272
10. NOTES AND COMMENTS 274
CHAPTER X. SOME CLASSICAL INTERPOLATION PROBLEMS 275
1. THE EIGENSPACE FOR S(m) 275
2. TANGENTIAL CARATHEODORY INTERPOLATION
REVISITED 284
3. TANGENTIAL NEVANLINNA PICK INTERPOLATION
REVISITED 290
4. THE TANGENTIAL HERMITE FEJER INTERPOLATION
PROBLEM 294
5. CLASSICAL HERMITE FEJER INTERPOLATION 298
6. MATRIX REPRESENTATIONS FOR FINITE RANK HANKEL
MATRICES 305
7. STATE SPACE REALIZATIONS 309
8. A HANKEL VERSION OF THE SCHUR COHN TEST 319
9. NOTES AND COMMENTS 325
CHAPTER XL INTERPOLATION AS A MOMENTUM PROBLEM 327
1. THE BASIC MOMENTUM RESULT 327
2. TANGENTIAL CARATHEODORY INTERPOLATION AS A
MOMENTUM PROBLEM 331
3. TANGENTIAL NEVANLINNA PICK INTERPOLATION AS A
MOMENTUM PROBLEM 333
4. TANGENTIAL HERMITE FEJER INTERPOLATION AS A
MOMENTUM PROBLEM 336
5. LOEWNER INTERPOLATION 339
xx TABLE OF CONTENTS
Page
6. NOTES AND COMMENTS 342
CHAPTER Xn. NUMERICAL ALGORITHMS FOR H~
OPTIMIZATION IN CONTROL THEORY 343
1. AN INTRODUCTION TO H°° CONTROL THEORY 343
2. MINIMAL FUNCTIONS AND SPECTRUM 352
3. A NUMERICAL ALGORITHM FOR H~ OPTIMIZATION 359
4. NOTES AND COMMENTS 365
CHAPTER XHI. INVERSE SCATTERING ALGORITHMS FOR THE
COMMUTANT LIFTING THEOREM 367
1. A PARAMETERIZATION OF BLOCK MATRIX
CONTRACTIONS 368
2. CHOICE SEQUENCES AND THE COMMUTANT LIFTING
THEOREM 374
3. SCHUR REPRESENTATIONS FOR THE COMMUTANT LIFTING
THEOREM 385
4. OBTAINING THE CHOICE SEQUENCE FROM 2 by 2 MATRIX
EXTENSIONS 394
5. TWO INVERSE SCATTERING ALGORITHMS FOR THE
COMMUTANT LIFTING THEOREM 400
6. TWO MORE INVERSE SCATTERING ALGORITHMS 404
7. OPERATOR QUASI BALLS 408
8. A PARAMETERIZATION OF ALL CONTRACTIVE
INTERTWINING LIFTINGS FOR A STRICTLY
CONTRACTIVE HANKEL OPERATOR 410
9. COMPUTING ALL CONTRACTIVE INTERTWINING LIFTINGS
FOR RATIONAL HANKEL OPERATORS 419
10. NOTES AND COMMENTS 425
TABLE OF CONTENTS xxi
Page
CHAPTER XIV. THE SCHUR REPRESENTATION 427
1. REDHEFFER PRODUCTS 427
2. REDHEFFER PRODUCTS AND THE STRUCTURE OF 2 by 2
MATRIX CONTRACTIONS 438
3. REDHEFFER PRODUCTS AND CARATHEODORY
INTERPOLATION 445
4. A REDHEFFER PRODUCT APPROACH TO THE SCHUR
REPRESENTATION 452
5. MORE ON THE SCHUR REPRESENTATION 460
6. ANOTHER REDHEFFER CASCADING INTERPRETATION OF
THE COMMUTANT LIFTING THEOREM 466
7. REDHEFFER CASCADING AND THE COMPUTATION OF ALL
CONTRACTIVE INTERTWINING LIFTINGS 471
8. A COMPUTATIONAL PROCEDURE FOR HERMITE FEJER
INTERPOLATION 478
9. THE HANKEL CASE 486
10. NOTES AND COMMENTS 493
CHAPTER XV. A GEOMETRIC APPROACH TO POSITIVE
DEFINITE SEQUENCES 497
1. THE NAIMARK DILATION THEOREM 497
2. POSITIVE DEFINITE FUNCTIONS AND CHOICE
SEQUENCES 502
3. THE CARATHEODORY INTERPOLATION PROBLEM FOR
POSITIVE DEFINITE FUNCTIONS 505
4. AN INVERSE SCATTERING ALGORITHM 510
5. DUALITY AND CHOICE SEQUENCES 515
xxi i TABLE OF CONTENTS
Page
6. THE LEVINSON ALGORITHM FOR BLOCK TOEPLITZ
MATRICES 517
7. THE NAIMARK DILATION AND MARINE SEISMOLOGY 522
8. THE INVERSE CAYLEY TRANSFORM OF Rn.... 526
9. THE NAIMARK DILATION FOR Hf (H, H) 531
10. THE NAIMARK DILATION AND LAYERED MEDIUM 536
11. THE LAYER PEELING ALGORITHM REVISITED 539
12. CHOICE SEQUENCES AND SCHUR NUMBERS 542
13. NOTES AND COMMENTS 545
CHAPTER XVI. POSITIVE DEFINITE BLOCK MATRICES 547
1. POSITIVE 2 by 2 BLOCK MATRICES 547
2. LEVINSON SYSTEMS 550
3. POSITIVE 3 by 3 BLOCK MATRICES 559
4. LEVINSON SYSTEMS FOR 3 by 3 BLOCK MATRICES 561
5. MORE RESULTS ON POSITIVE 3 by 3 BLOCK MATRICES 566
6. SOME CLASSICAL RESULTS FOR TOEPLITZ MATRICES
ON Cn REVISITED 567
7. POSITIVE n by n BLOCK MATRICES 570
8. DUALITY AND POSITIVE n by n BLOCK MATRICES 573
9. CHOICE SEQUENCES AND POSITIVE BLOCK TOEPLITZ
MATRICES REVISITED 576
10. DUALITY AND CHOICE SEQUENCES FOR POSITIVE
BLOCK TOEPLITZ MATRICES 580
11. NOTES AND COMMENTS 584
TABLE OF CONTENTS xxi i i
Page
CHAPTER XVH. A PHYSICAL BASIS FOR THE LAYERED
MEDIUM MODEL 587
1. WAVES IN AN ELASTIC MEDIUM 587
2. WAVES IN A LAYERED MEDIUM 594
3. NOTES AND COMMENTS 598
REFERENCES 599
NOTATION 625
INDEX 629
|
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| language | English |
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| physical | XXIII, 632 S. |
| publishDate | 1990 |
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| series | Operator theory |
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| spelling | Foiaş, Ciprian 1933-2020 Verfasser (DE-588)172077648 aut The commutant lifting approach to interpolation problems Ciprian Foias ; Arthur E. Frazho Basel u.a. Birkhäuser 1990 XXIII, 632 S. txt rdacontent n rdamedia nc rdacarrier Operator theory 44 Interpolation Lifting theory Liftungssatz (DE-588)4265072-0 gnd rswk-swf Interpolation (DE-588)4162121-9 gnd rswk-swf Kommutant (DE-588)4265073-2 gnd rswk-swf Liften Mathematik (DE-588)4167655-5 gnd rswk-swf Liften Mathematik (DE-588)4167655-5 s Interpolation (DE-588)4162121-9 s DE-604 Kommutant (DE-588)4265073-2 s Liftungssatz (DE-588)4265072-0 s Frazho, Arthur E. Verfasser aut Operator theory 44 (DE-604)BV000000970 44 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=001675589&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Foiaş, Ciprian 1933-2020 Frazho, Arthur E. The commutant lifting approach to interpolation problems Operator theory Interpolation Lifting theory Liftungssatz (DE-588)4265072-0 gnd Interpolation (DE-588)4162121-9 gnd Kommutant (DE-588)4265073-2 gnd Liften Mathematik (DE-588)4167655-5 gnd |
| subject_GND | (DE-588)4265072-0 (DE-588)4162121-9 (DE-588)4265073-2 (DE-588)4167655-5 |
| title | The commutant lifting approach to interpolation problems |
| title_auth | The commutant lifting approach to interpolation problems |
| title_exact_search | The commutant lifting approach to interpolation problems |
| title_full | The commutant lifting approach to interpolation problems Ciprian Foias ; Arthur E. Frazho |
| title_fullStr | The commutant lifting approach to interpolation problems Ciprian Foias ; Arthur E. Frazho |
| title_full_unstemmed | The commutant lifting approach to interpolation problems Ciprian Foias ; Arthur E. Frazho |
| title_short | The commutant lifting approach to interpolation problems |
| title_sort | the commutant lifting approach to interpolation problems |
| topic | Interpolation Lifting theory Liftungssatz (DE-588)4265072-0 gnd Interpolation (DE-588)4162121-9 gnd Kommutant (DE-588)4265073-2 gnd Liften Mathematik (DE-588)4167655-5 gnd |
| topic_facet | Interpolation Lifting theory Liftungssatz Kommutant Liften Mathematik |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=001675589&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000000970 |
| work_keys_str_mv | AT foiasciprian thecommutantliftingapproachtointerpolationproblems AT frazhoarthure thecommutantliftingapproachtointerpolationproblems |