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Chebyshev Polynomials: from Approximation Theory to Algebra and Number Theory: Second Edition [Minkštas viršelis]

  • Formatas: Paperback / softback, 272 pages, aukštis x plotis x storis: 10x150x225 mm, weight: 390 g
  • Išleidimo metai: 30-Sep-2020
  • Leidėjas: Dover Publications Inc.
  • ISBN-10: 0486842339
  • ISBN-13: 9780486842332
Kitos knygos pagal šią temą:
Chebyshev Polynomials: from Approximation Theory to Algebra and Number Theory: Second EditionDidesnis paveikslėlis
  • Formatas: Paperback / softback, 272 pages, aukštis x plotis x storis: 10x150x225 mm, weight: 390 g
  • Išleidimo metai: 30-Sep-2020
  • Leidėjas: Dover Publications Inc.
  • ISBN-10: 0486842339
  • ISBN-13: 9780486842332
Kitos knygos pagal šią temą:
Pastovi nuoroda: https://www.kriso.lt/db/9780486842332.html
Raktažodžiai:
"This survey of the most important properties of the Chebyshev polynomials encompasses several areas of mathematical analysis: interpolation theory, orthogonal polynomials, approximation theory, numerical integration, numerical analysis, and ergodic theory. Originally published in 1974, the text was updated in 1990; this reprint of the second edition corrects various errors and features new material, including a chapter introducing elementary algebraic and number theoretic properties of Chebyshev polynomials"--

This survey of the most important properties of the Chebyshev polynomials encompasses several areas of mathematical analysis: interpolation theory, orthogonal polynomials, approximation theory, numerical integration, numerical analysis, and ergodic theory. 1990 edition.


This survey of the most important properties of Chebyshev polynomials encompasses several areas of mathematical analysis:
&; Interpolation theory
&; Orthogonal polynomials
&; Approximation theory
&; Numerical integration
&; Numerical analysis
&; Ergodic theory
Starting with some definitions and descriptions of elementary properties, the treatment advances to examinations of extremal properties, the expansion of functions in a series of Chebyshev polynomials, and iterative properties. The final chapter explores selected algebraic and number theoretic properties of the Chebyshev polynomials.
For advanced undergraduates and graduate students in mathematics
Originally published in 1974, the text was updated in 1990; this reprint of the second edition corrects various errors and features new material.
1 Definitions And Some Elementary Properties
1(56)
1.1 Definition of the Chebyshev Polynomials
1(4)
Exercises 1.1.1--1.1.6
5(1)
1.2 Some Simple Properties
5(5)
Exercises 1.2.1--1.2.23
7(3)
1.3 Polynomial Interpolation at the Zeros and Extrema
10(18)
Exercises 1.3.1--1.3.24
23(5)
1.4 Hermite Interpolation
28(6)
Exercises 1.4.1--1.4.10
29(5)
Exercises 1.4.11--1.4.12
34(1)
1.5 Orthogonality
34(23)
1 Second Order Linear Homogeneous Differential Equation
36(1)
Exercises 1.5.1--1.5.13
37(2)
2 Three-Term Recurrence Formula
39(1)
Exercises 1.5.14---1.5.19
40(1)
3 Generating Function
41(1)
4 Least Squares
42(1)
5 Numerical Integration
43(3)
Exercises 1.5.20--1.5.25
46(7)
Exercises 1.5.26--1.5.67
53(4)
2 Extremal Properties
57(98)
A Uniform Approximation of Continuous Functions
68(29)
2.1 Convex Sets in n-Space
68(3)
Exercises 2.1.1-2.1.5
71(1)
2.2 Characterization of Best Approximations
71(5)
Exercises 2.2.1-2.2.15
76(2)
2.3 Chebyshcv Systems and Uniqueness
78(5)
Exercises 2.3.1--2.3.4
83(1)
2.4 Approximation on an Interval
84(4)
Exercises 2.4.1--2.4.50
88(9)
B Maximizing Linear Functional on n
97(58)
2.5 An Interpolation Formula for Linear Functionals
97(2)
Exercises 2.5.1--2.5.12
99(3)
2.6 Linear Functionals on n
102(4)
Exercises 2.6.1--2.6.13
106(1)
2.7 Some Examples in which the Chebyshev Polynomials Are Extremal
107(1)
1 Growth Outside the Interval
108(2)
2 Size of Coefficients
110(3)
3 The Tau Method
113(5)
4 Size of the Derivative
118(5)
5 V. A. Markov's Theorem
123(15)
Exercises 2.7.1--2.7.14
138(3)
2.8 Additional Extremal Problems
141(1)
1 More About the Bernstein and Markov Inequalities
141(1)
1.1 Polynomial Inequalities in the Complex Plane
141(4)
1.2 Polynomials with Curved Majorants
145(2)
Exercises 2.8.1--2.8.8
147(2)
2 Miscellaneous Extremal Properties
149(1)
2.1 The Remez Inequality for Polynomials
149(1)
2.2 The Longest Polynomial
149(2)
2.3 An Iterative Solution of a System of Linear Equations
151(4)
3 Expansion Of Functions In Series Of Chebyshev Polynomials
155(37)
3.1 Polynomials in Chebyshev Form
155(1)
3.2 Evaluating Polynomials in Chebyshev Form
156(5)
Exercises 3.2.1--3.2.5
159(2)
3.3 Chebyshev Series
161(5)
3.4 The Relationship of Sn to En
166(14)
Exercises 3.4.1--3.4.7
168(11)
Exercises 3.4.8--3.4.12
179(1)
3.5 The Evaluation of Chebyshev Coefficients
180(8)
Exercises 3.5.1--3.5.4
187(1)
3.6 An Optimal Property of Chebyshev Expansions
188(4)
4 Iterative Properties And Some Remarks About The Graphs Of The Tn
192(25)
4.1 Permutable Polynomials
192(8)
Exercises 4.1.1--4.1.9
196(4)
4.2 Ergodic and Mixing Properties
200(8)
4.3 The "White" Curves and Intersection Points of Pairs of Chebyshev Polynomials
208(9)
5 Some Algebraic And Number Theoretic Properties Of The Chebyshev Polynomials
217(17)
5.1 The Discriminant of the Chebyshev Polynomials
217(3)
Exercises 5.1.1.--5.1.4
219(1)
5.2 The Factorization of the Chebyshev Polynomials into Polynomials with Rational Coefficients
220(11)
1 Preliminary Definitions and Results
220(1)
Exercises 5.2.1.--5.2.23
221(3)
2 The Irreducibility of the Cyclotomic Polynomials
224(2)
Exercises 5.2.24--5.2.25
226(1)
3 The Factorization of the Chebyshev Polynomials Over Q
227(3)
Exercises 5.2.26--5.2.29
230(1)
5.3 Some Number Theoretic Properties of the Chebyshev Polynomials
231(3)
1 Pell's Equation
231(1)
2 Fermat's Theorem for the Chebyshev Polynomials
232(1)
3 (n(x), m(x)) = (m,n)(x)
232(2)
References 234(10)
Glossary of Symbols 244(3)
Index 247
Theodore J. Rivlin (1926-2006) was on the staff of the IBM Research Division, Thomas J. Watson Research Center, Yorktown Heights, New York. His other Dover book is An Introduction to the Approximation of Functions.