Interpolation and approximation by rational functions in the complex domain
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| 1. Verfasser: | |
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| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Providence, RI
American Math. Soc.
1969
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| Ausgabe: | 5. ed. |
| Schriftenreihe: | Colloquium publications
20 |
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| 245 | 1 | 0 | |a Interpolation and approximation by rational functions in the complex domain |c by J. L. Walsh |
| 250 | |a 5. ed. | ||
| 264 | 1 | |a Providence, RI |b American Math. Soc. |c 1969 | |
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| adam_text | AMERICAN MATHEMATICAL SOCIETY COLLOQUIUM PUBLICATIONS VOLUME XX
INTERPOLATION AND APPROXIMATION BY RATIONAL FUNCTIONS IN THE COMPLEX
DOMAIN BY J. L. WALSH PUBLISHED BY THE AMERICAN MATHEMATICAL SOCIETY
PROVIDENCE, RHODE ISLAND TABLE OF CONTENTS PAGE PREFACE III CHAPTER I
POSSIBILITY OP APPROXIMATION; ANALYTIC FUNCTIONS $1.1. POINT SETS:
PRELIMINARY DEFINITIONS 1 §1.2. FUNCTION-THEORETIC CONSIDERATIONS 4
§1.3. AN OPEN SET AS THE SUM OF REGIONS 6 §1.4. EXPANSION OF AN ANALYTIC
FUNCTION 10 §1.5. A THEOREM ON ANALYTIC EXTENSION. 12 §1.6.
APPROXIMATION; CHOICE OF POLES 14 §1.7. COMPONENTS OF AN ANALYTIC
FUNCTION 17 §1.8. METHODS OF APPELL AND OF WOLFF 19 §1.9. ON THE
VANISHING OF ANALYTIC FUNCTIONS 22 §1.10. NECESSARY CONDITIONS FOR
APPROXIMATION 23 CHAPTER II POSSIBILITY OF APPROXIMATION, CONTINUED
§2.1. LINDELOF S FIRST THEOREM 27 §2.2. LINDELOF S SECOND THEOREM 29
§2.3. CONFORMAL MAPPING OF VARIABLE REGIONS 32 §2.4. APPROXIMATION IN A
CLOSED JORDAN REGION 36 §2.5. APPLICATIONS, JORDAN CONFIGURATIONS 39
§2.6. GENERAL FORMS OF CAUCHY S INTEGRAL FORMULA 42 §2.7. SURFACE
INTEGRALS AS MEASURES OF APPROXIMATION 44 §2.8. UNIFORM APPROXIMATION;
FURTHER RESULTS 46 CHAPTER III INTERPOLATION AND LEMNISCATES §3.1.
POLYNOMIALS OF INTERPOLATION 49 §3.2. SEQUENCES AND SERIES OF
INTERPOLATION 52 §3.3. LEMNISCATES AND THE JACOBI SERIES 54 §3.4. AN
ANALOGOUS SERIES OF INTERPOLATION 56 §3.5. A MORE GENERAL SERIES OF
INTERPOLATION 60 CHAPTER IV DEGREE OF CONVERGENCE OF POLYNOMIALS.
OVERSONVERGENCE §4.1. EQUIPOTENTIAL CURVES IN CONFORMAL MAPS 65 §4.2.
APPROXIMATION OF JORDAN CURVES BY A LEMNISCATE 68 §4.3. APPROXIMATION OF
MODULUS OF THE MAPPING FUNCTION 71 §4.4. APPROXIMATION OF MODULUS OF
MAPPING FUNCTION, CONTINUED 74 §4.5. DEGREE OF CONVERGENCE. SUFFICIENT
CONDITIONS 75 §4.6. DEGREE OF CONVERGENCE. NECESSARY CONDITIONS.
OVERCONVERGENCE 77 §4.7. MAXIMAL CONVERGENCE 79 §4.8. EXACT REGIONS OF
UNIFORM CONVERGENCE 88 §4.9. APPROXIMATION ON MORE GENERAL POINT SETS
(IRREGULAR CASE) 85 VII VLLL TABLE OF CONTENTS CHAPTER V BEST
APPROXIMATION BY POLYNOMIALS §5.1. TCHEBYEHEFF APPROXIMATION 89 §5.2.
APPROXIMATION MEASURED BY A LINE INTEGRAL 91 §5.3. APPROXIMATION
MEASURED BY A SURFACE INTEGRAL 95 §5.4. APPROXIMATION MEASURED BY A LINE
INTEGRAL AFTER CONFORMAL MAPPING OF COM- PLEMENT ,98 §5.5. APPROXIMATION
MEASURED BY A LINE INTEGRAL AFTER CONFORMAL MAPPING OF INTERIOR. 100
§5.6. POINT SETS WITH INFINITELY MANY COMPONENTS 102 §5.7. GENERALITY OF
WEIGHT FUNCTIONS 104 §5.8. APPROXIMATION OF FUNCTIONS NOT ANALYTIC ON
CLOSED SET CONSIDERED 107 CHAPTER VI ORTHOGONALITY AND LEAST SQUARES
§6.1. ORTHOGONAL FUNCTIONS AND LEAST SQUARES ILL §6.2. ORTHOGONALIZATION
113 §6.3. RIESZ-FISCHER THEORY. 116 §6.4. CLOSURE 120 §6.5. POLYNOMIAL
APPROXIMATION TO ANALYTIC FUNCTIONS 125 §6.6. ASYMPTOTIC PROPERTIES OF
COEFFICIENTS 128 §6.7. REGIONS OF CONVERGENCE 131 §6.8. POLYNOMIALS
ORTHOGONAL ON SEVERAL CURVES 133 §6.9. FUNCTIONS OF THE SECOND KIND 136
§6.10. FUNCTIONS OF CLASS H 2 141 §6.11. POLYNOMIALS IN Z AND 1/Z 143
§6.12. AN EXTREMAL PROBLEM, LINE INTEGRALS 146 §6.13. AN EXTREMAL
PROBLEM, SURFACE INTEGRALS 149 CHAPTER VII INTERPOLATION BY POLYNOMIALS
§7.1. INTERPOLATION IN ROOTS OF UNITY 152 §7.2. A SUFFICIENT CONDITION
FOR MAXIMAL CONVERGENCE 154 §7.3. A NECESSARY CONDITION FOR UNIFORM
CONVERGENCE 159 §7.4. FURTHER CONDITIONS FOR MAXIMAL CONVERGENCE 162
§7.5. UNIFORM DISTRIBUTION OF POINTS 164 §7.6. INTERPOLATION IN POINTS
UNIFORMLY DISTRIBUTED 167 §7.7. POINTS OF INTERPOLATION WITH EXTREMAL
PROPERTIES 170 §7.8. EXISTENCE OF POLYNOMIALS CONVERGING MAXIMALLY
(SHEN) 173 §7.9. A SYNTHESIS OF INTERPOLATION AND TCHEBYCHEFF
APPROXIMATION 175 §7.10. LEAST SQUARES AND INTERPOLATION IN ROOTS OF
UNITY 178 CHAPTER VIII & INTERPOLATION BY RATIONAL FUNCTIONS §8.1.
INTERPOLATION FORMULAS 184 §8.2. SEQUENCES AND SERIES OF INTERPOLATION
188 §8.3. DUALITY: GENERAL THEOREMS 193 §8.4. DUALITY: ILLUSTRATIONS 199
§8.5. DUALITY AND SERIES OF INTERPOLATION 203 §8.6. ILLUSTRATIONS 206
§8.7. HARMONIC FUNCTIONS AS GENERATING FUNCTIONS 209 §8.8. HARMONIC
FUNCTIONS AS GENERATING FUNCTIONS, CONTINUED 212 TABLE OF CONTENTS IX
§8.9. GEOMETRIC CONDITIONS ON GIVEN POINTS 218 §8.10. GEOMETRIC
CONDITIONS, CONTINUATION 221 CHAPTER IX APPROXIMATION BY RATIONAL
FUNCTIONS §9.1. LEAST SQUARES ON THE UNIT CIRCLE AND INTERPOLATION 224
§9.2. UNIT CIRCLE. CONVERGENCE THEOREMS 228 §9.3. UNIT CIRCLE. OTHER
MEASURES OF APPROXIMATION 231 §9.4. UNIT CIRCLE. ASYMPTOTIC CONDITIONS
ON POLES 235 §9.5. APPLICATIONS 239 §9.6. POLES WITH LIMIT POINTS ON
CIRCUMFERENCE 243 §9.7. GENERAL POINT SETS; DEGREE OF CONVERGENCE 249
§9.8. GENERAL POINT SETS; BEST APPROXIMATION 252 §9.9. EXTENSIONS 257
§9.10. GENERAL POINT SETS; ASYMPTOTIC CONDITIONS ON POLES 261 §9.11.
OPERATIONS WITH ASYMPTOTIC CONDITIONS 265 §9.12. ASYMPTOTIC CONDITIONS
UNDER CONFORMAL TRANSFORMATION 270 §9.13. FURTHER PROBLEMS 276 CHAPTER X
INTERPOLATION AND FUNCTIONS ANALYTIC IN THE UNIT CIRCLE §10.1. THE
BLASCHKE PRODUCT 281 §10.2. FUNCTIONS OF MODULUS NOT GREATER THAN M 286
§10.3. FUNCTIONS OF LEAST MAXIMUM MODULUS 290 §10.4. CONVERGENCE OF
MINIMIZING SEQUENCES 293 §10.5. TOTALITY OF INTERPOLATING FUNCTIONS 296
§10.6. CONDITIONS FOR UNIQUENESS 300 §10.7. FUNCTIONS OF CLASS H 2 304
CHAPTER XI APPROXIMATION WITH AUXILIARY CONDITIONS AND TO NON-ANALYTIC
FUNCTIONS §11.1. APPROXIMATION WITH INTERPOLATION TO GIVEN FUNCTION 310
§11.2. INTERPOLATION TO GIVEN FUNCTION; DEGREE OF CONVERGENCE 314 §11.3.
EXTREMAL PROBLEMS INVOLVING AUXILIARY CONDITIONS 318 §11.4. EXPANSION OF
MAPPING FUNCTION 322 §11.5. APPROXIMATION ON A RECTIFIABLE JORDAN CURVE
328 §11.6. INTERPOLATION IN ROOTS OF UNITY 333 §11.7. TCHEBYCHEFF
MEASURE OF APPROXIMATION; EXTREMAL PROBLEMS 335 §11.8. TCHEBYCHEFF
APPROXIMATION BY POLYNOMIALS AND RATIONAL FUNCTIONS 338 §11.9.
APPROXIMATION BY NON-VANISHING FUNCTIONS 343 CHAPTER XII C EXISTENCE AND
UNIQUENESS OF RATIONAL FUNCTIONS OF BEST APPROXIMATION §12.1. SEQUENCES
OF RATIONAL FUNCTIONS OF GIVEN DEGREE 348 §12.2. APPLICATION TO
TCHEBYCHEFF APPROXIMATION 351 §12.3. RESTRICTION OF LOCATION OF POLES
353 §12.4. THE RATIONAL FUNCTION OF BEST APPROXIMATION NEED NOT BE
UNIQUE 356 §12.5. INTEGRAL MEASURES OF APPROXIMATION 357 §12.6.
APPROXIMATION WITH AUXILIARY CONDITIONS 360 §12.7. UNIQUENESS OF
APPROXIMATING FUNCTIONS WITH PREASSIGNED POLES 361 X TABLE OF CONTENTS
APPENDIX §A1. POSSIBILITY OF APPROXIMATION OF POLYNOMIALS 367 §A2.
APPROXIMATION BY POLYNOMIALS. CONTINUITY CONDITIONS 371 §A3.
INTERPOLATION AND APPROXIMATION BY BOUNDED ANALYTIC FUNCTIONS 373 §A4.
THE CONVERGENCE OF SEQUENCES OF RATIONAL FUNCTIONS OF BEST APPROXIMATION
WITH SOME FREE POLES 378 BIBLIOGRAPHY . * 383 INDEX 397
|
| any_adam_object | 1 |
| author | Walsh, Joseph L. 1895-1973 |
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| author_facet | Walsh, Joseph L. 1895-1973 |
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| dewey-full | 517.21 |
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| dewey-raw | 517.21 |
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| discipline | Mathematik |
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| id | DE-604.BV006247661 |
| illustrated | Not Illustrated |
| indexdate | 2024-12-23T12:03:43Z |
| institution | BVB |
| isbn | 0821810200 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-003945148 |
| oclc_num | 950945 |
| open_access_boolean | |
| owner | DE-739 DE-355 DE-BY-UBR DE-824 DE-384 DE-706 |
| owner_facet | DE-739 DE-355 DE-BY-UBR DE-824 DE-384 DE-706 |
| physical | X, 405 S. |
| publishDate | 1969 |
| publishDateSearch | 1969 |
| publishDateSort | 1969 |
| publisher | American Math. Soc. |
| record_format | marc |
| series | Colloquium publications |
| series2 | Colloquium publications |
| spellingShingle | Walsh, Joseph L. 1895-1973 Interpolation and approximation by rational functions in the complex domain Colloquium publications Funcoes Algebricas larpcal Functions Interpolation Series, Infinite Approximationstheorie (DE-588)4120913-8 gnd Komplexität (DE-588)4135369-9 gnd Rationale Funktion (DE-588)4225679-3 gnd Approximation (DE-588)4002498-2 gnd Interpolation (DE-588)4162121-9 gnd |
| subject_GND | (DE-588)4120913-8 (DE-588)4135369-9 (DE-588)4225679-3 (DE-588)4002498-2 (DE-588)4162121-9 |
| title | Interpolation and approximation by rational functions in the complex domain |
| title_auth | Interpolation and approximation by rational functions in the complex domain |
| title_exact_search | Interpolation and approximation by rational functions in the complex domain |
| title_full | Interpolation and approximation by rational functions in the complex domain by J. L. Walsh |
| title_fullStr | Interpolation and approximation by rational functions in the complex domain by J. L. Walsh |
| title_full_unstemmed | Interpolation and approximation by rational functions in the complex domain by J. L. Walsh |
| title_short | Interpolation and approximation by rational functions in the complex domain |
| title_sort | interpolation and approximation by rational functions in the complex domain |
| topic | Funcoes Algebricas larpcal Functions Interpolation Series, Infinite Approximationstheorie (DE-588)4120913-8 gnd Komplexität (DE-588)4135369-9 gnd Rationale Funktion (DE-588)4225679-3 gnd Approximation (DE-588)4002498-2 gnd Interpolation (DE-588)4162121-9 gnd |
| topic_facet | Funcoes Algebricas Functions Interpolation Series, Infinite Approximationstheorie Komplexität Rationale Funktion Approximation |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=003945148&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV035417609 |
| work_keys_str_mv | AT walshjosephl interpolationandapproximationbyrationalfunctionsinthecomplexdomain |