Chebyshev polynomials

✍ Scribed by Theodore J. Rivlin

• Publisher: Wiley
• Year: 1990
• Tongue: English
• Leaves: 268
• Series: Pure and Applied Mathematics
• Edition: 2
• Category: Library

✦ Table of Contents

1 Definitions and Some Elementary Properties
1.1 Definition of the Chebyshev Polynomials, 1
Exercises 11.1—11.6, 5
1.2 Some Simple Properties, 5
Exercises [2.1—1.2.23, 7
1.3 Polynomial Interpolation at the Zeros and
Extrema, 10
Exercises 1.3.1 —I 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-Tenn Recurrence Formula, 39
Exercises I.5.I4—I.5.19, 40
3. Generating Function, 41
4. Least Squares, 42
5. Numerical Integration, 43
Exercises 15.20—15.25, 46
Exercises 1 .5.26-I .5 .6 7, 53
2 Extremal Properties
A Uniform Approximation of Continuous Functions, 68
2.1 Convex Sets in n-Space, 68
Exercises 21.1—21.5, 71
2.2 Characterization of Best Approximations, 71
Exercises 2.2.1-2.2.15, 76
2.3 Chebyshev Systems and Uniqueness, 78
Exercises 23.1—23.4, 83
2.4 Approximation on an Interval, 84
Exercises 2.4.1—2.4.50, 88
B Maximizing Linear Functionals on 9,, 97'
2.5 An Interpolation Formula for Linear Functionals, 97
Exercises 2.5.1—2.5.12, 99
2.6 Linear Functionals on a" 102
Exercises 2.6.1—2.6.I3, 106
2.7 Some Examples in which the Chebyshev Polynomials
Are Extremal, 107
1. Growth Outside the Interval, 108
Size Of Coefficients, 110‘
The Tau Method, 113
Size of the Derivative, 118
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 INEQUALmEs IN THE COMPLEX
PLANE, 141
1.2 POLYNOMIALS WITH CURVED
MAJORAN'IS, 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 S, to E”, 166
Exercises 3.4.1—3.4.7, 168
Exercises 3.4.8—3.4.12, I79
3.5 The Evaluation of Chebyshev Coeflicients, 180
Exercises 3.51-3.54, 187
3.6 An Optimal Property of Chebyshev Expansions, 188
4 Iterative Properties and Some Remarks About the Graphs of
the I; 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 Coeflicients, 220
1. Preliminary Definitions and Results, 220
Exercises 5.2.1.—5.2.23, 221
2. The Irreducibility of the Cyclotomic
Polynomials, 224
Exercises 52.24—52.25, 226
3. The Factorization of the Chebyshev Polynomials
Over 0, 227
Exercises 52.26—52.29, 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' (%,(X), %m(x)) = W(m.u)(x)9 232
References 234
Glossary of Symbols 244
Index