Chebyshev polynomials: from approximation theory to algebra and number theory
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New York [u.a.]
Wiley
1990
|
| Edition: | 2. ed. |
| Series: | Pure and applied mathematics
A Wiley-Interscience publication |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XIII, 249 S. |
| ISBN: | 0471628964 |
Staff View
MARC
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Record in the Search Index
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| adam_text | CONTENTS
1 Definitions and Some Elementary Properties 1
1.1 Definition of the Chebyshev Polynomials, 1
Exercises 1.1.1 1.1.6, 5
1.2 Some Simple Properties, 5
Exercises 1.2.1 1.2.23, 7
1.3 Polynomial Interpolation at the Zeros and
Extrema, 10
Exercises 13.1 1.3.24, 23
1.4 Hermite Interpolation, 28
Exercises 1.4.1 1.4.10, 29
Exercises 1.4.11 1.4.12, 34
1.5 Orthogonality, 34
1. Second Order Linear Homogeneous Differential
Equation, 36
Exercises 1.5.1 1.5.13, 37
2. Three Term Recurrence Formula, 39
Exercises 1.5.14 1.5.19, 40
3. Generating Function, 41
4. Least Squares, 42
5. Numerical Integration, 43
Exercises 1.5.20 1.5.25, 46
Exercises 1.5.26 1.5.67, 53
2 Extremal Properties 67
A Uniform Approximation of Continuous Functions, 68
2.1 Convex Sets in n Space, 68
Exercises 2.1.1 2.1.5, 71
2.2 Characterization of Best Approximations, 71
Exercises 2.2.1 2.2.15, 76
xii CONTENTS
2.3 Chebyshev Systems and Uniqueness, 78
Exercises 2.3.1 2.3.4, 83
2.4 Approximation on an Interval, 84
Exercises 2.4.1 2.4.50, 88
B Maximizing Linear Functionals on „ 97
2.5 An Interpolation Formula for Linear Functionals, 97
Exercises 2.5.1 2.5.12, 99
2.6 Linear Functionals on 3?n, 102
Exercises 2.6.1 2.6.13, 106
2.7 Some Examples in which the Chebyshev Polynomials
Are Extremal, 107
1. Growth Outside the Interval, 108
2. Size of Coefficients, 110
3. TheTau Method, 113
4. Size of the Derivative, 118
5. V. A. Markov s Theorem, 123
Exercises 2.7.1 2.7.14, 138
2.8 Additional Extremal Problems, 141
1. More About the Bernstein and Markov
Inequalities, 141
1.1 Polynomial Inequalities in the Complex
Plane, 141
1.2 Polynomials with Curved
Majorants, 145
Exercises 2.8.1 2.8.8, 147
2. Miscellaneous Extremal Properties, 149
2.1 The Remez Inequality for
Polynomials, 149
2.2 The Longest Polynomial, 149
2.3 An Iterative Solution of a System of
Linear Equations, 151
3 Expansion of Functions in Series of Chebyshev Polynomials 155
3.1 Polynomials in Chebyshev Form, 155
3.2 Evaluating Polynomials in Chebyshev Form, 156
Exercises 3.2.1 3.2.5, 159
3.3 Chebyshev Series, 161
3.4 The Relationship of Sn to En, 166
Exercises 3.4.1 3.4.7, 168
Exercises 3.4.8 3.4.12, 179
CONTENTS xiii
3.5 The Evaluation of Chebyshev Coefficients, 180
Exercises 3.5.1 3.5.4, 187
3.6 An Optimal Property of Chebyshev Expansions, 188
4 Iterative Properties and Some Remarks About the Graphs of
the TH 192
4.1 Permutable Polynomials, 192
Exercises 4.1.1 4.1.9, 196
4.2 Ergodic and Mixing Properties, 200
4.3 The White Curves and Intersection Points of Pairs of
Chebyshev Polynomials, 208
5 Some Algebraic and Number Theoretic Properties of the
Chebyshev Polynomials 217
5.1 The Discriminant of the Chebyshev Polynomials, 217
Exercises 5.1.1. 5.1.4, 219
5.2 The Factorization of the Chebyshev Polynomials into
Polynomials with Rational Coefficients, 220
1. Preliminary Definitions and Results, 220
Exercises 5.2.1. 5.2.23, 221
2. The Irreducibility of the Cyclotomic
Polynomials, 224
Exercises 5.2.24 5.2.25, 226
3. The Factorization of the Chebyshev Polynomials
Over Q, 227
Exercises 5.2.26 5.2.29, 230
5.3 Some Number Theoretic Properties of the Chebyshev
Polynomials, 231
1. Pell s Equation, 231
2. Fermat s Theorem for the Chebyshev
Polynomials, 232
3. (?„(*), «„(*)) = %mJx), 232
References 234
Glossary of Symbols 244
Index 247
|
| any_adam_object | 1 |
| author | Rivlin, Theodore J. |
| author_facet | Rivlin, Theodore J. |
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| author_sort | Rivlin, Theodore J. |
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| callnumber-subject | QA - Mathematics |
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| classification_tum | MAT 412f MAT 338f |
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| dewey-full | 515/.55 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 515 - Analysis |
| dewey-raw | 515/.55 |
| dewey-search | 515/.55 |
| dewey-sort | 3515 255 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| edition | 2. ed. |
| format | Book |
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| isbn | 0471628964 |
| language | English |
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| physical | XIII, 249 S. |
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| series2 | Pure and applied mathematics A Wiley-Interscience publication |
| spelling | Rivlin, Theodore J. Verfasser aut Chebyshev polynomials from approximation theory to algebra and number theory Theodore J. Rivlin 2. ed. New York [u.a.] Wiley 1990 XIII, 249 S. txt rdacontent n rdamedia nc rdacarrier Pure and applied mathematics A Wiley-Interscience publication Tchebychev, polynômes de ram approximation inriac meilleure approximation inriac polynôme Chebyshev inriac Chebyshev polynomials Approximationstheorie (DE-588)4120913-8 gnd rswk-swf Čebyšev-Polynome (DE-588)4147437-5 gnd rswk-swf Polynom (DE-588)4046711-9 gnd rswk-swf Čebyšev-Polynome (DE-588)4147437-5 s DE-604 Approximationstheorie (DE-588)4120913-8 s Polynom (DE-588)4046711-9 s 1\p DE-604 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=002628632&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Rivlin, Theodore J. Chebyshev polynomials from approximation theory to algebra and number theory Tchebychev, polynômes de ram approximation inriac meilleure approximation inriac polynôme Chebyshev inriac Chebyshev polynomials Approximationstheorie (DE-588)4120913-8 gnd Čebyšev-Polynome (DE-588)4147437-5 gnd Polynom (DE-588)4046711-9 gnd |
| subject_GND | (DE-588)4120913-8 (DE-588)4147437-5 (DE-588)4046711-9 |
| title | Chebyshev polynomials from approximation theory to algebra and number theory |
| title_auth | Chebyshev polynomials from approximation theory to algebra and number theory |
| title_exact_search | Chebyshev polynomials from approximation theory to algebra and number theory |
| title_full | Chebyshev polynomials from approximation theory to algebra and number theory Theodore J. Rivlin |
| title_fullStr | Chebyshev polynomials from approximation theory to algebra and number theory Theodore J. Rivlin |
| title_full_unstemmed | Chebyshev polynomials from approximation theory to algebra and number theory Theodore J. Rivlin |
| title_short | Chebyshev polynomials |
| title_sort | chebyshev polynomials from approximation theory to algebra and number theory |
| title_sub | from approximation theory to algebra and number theory |
| topic | Tchebychev, polynômes de ram approximation inriac meilleure approximation inriac polynôme Chebyshev inriac Chebyshev polynomials Approximationstheorie (DE-588)4120913-8 gnd Čebyšev-Polynome (DE-588)4147437-5 gnd Polynom (DE-588)4046711-9 gnd |
| topic_facet | Tchebychev, polynômes de approximation meilleure approximation polynôme Chebyshev Chebyshev polynomials Approximationstheorie Čebyšev-Polynome Polynom |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=002628632&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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