After an introduction to the geometry of polynomials and a discussion of refinements of the Fundamental Theorem of Algebra, the book turns to a consideration of various special polynomials. Chebyshev and Descartes systems are then introduced, and Müntz systems and rational systems are examined in detail. Subsequent chapters discuss denseness questions and the inequalities satisfied by polynomials and rational functions. Appendices on algorithms and computational concerns, on the interpolation theorem, and on orthogonality and irrationality round off the text. The book is self-contained and assumes at most a senior-undergraduate familiarity with real and complex analysis.
Author(s): Peter Borwein, Tamás Erdélyi
Series: Graduate Texts in Mathematics, 161
Edition: 1
Publisher: Springer
Year: 1995
Language: English
Commentary: Pages i-v missing
Pages: 487
POLYNOMIALS AND POLYNOMIAL INEQUALITIES......Page 1
Dedication......Page 2
Preface......Page 3
Acknowledgments......Page 4
Contents......Page 5
1.1 Polynomials and Rational Functions......Page 7
1.2 The Fundamental Theorem of Algebra......Page 17
1.3 Zeros of the Derivative......Page 24
2.1 Chebyshev Polynomials......Page 35
2.2 Orthogonal Functions......Page 47
2.3 Orthogonal Polynomials......Page 63
2.4 Polynomials with Nonnegative Coefficients......Page 85
Overview......Page 97
3.1 Chebyshev Systems......Page 98
3.2 Descartes Systems......Page 106
3.3 Chebyshev Polynomials in Chebyshev Spaces......Page 120
3.4 Müntz–Legendre Polynomials......Page 131
3.5 Chebyshev Polynomials in Rational Spaces......Page 145
4.1 Variations on the Weierstrass Theorem......Page 160
4.2 Müntz's Theorem......Page 177
4.3 Unbounded Bernstein Inequalities......Page 212
4.4 Müntz Rationals......Page 224
5.1 Classical Polynomial Inequalities......Page 233
5.2 Markov's Inequality for Higher Derivatives......Page 254
5.3 Inequalities for Norms of Factors......Page 266
6.1 Inequalities in Müntz Spaces......Page 281
6.2 Nondense Müntz Spaces......Page 309
7.1 Inequalities for Rational Function Spaces......Page 326
7.2 Inequalities for Logarithmic Derivatives......Page 350
Appendix A1. Algorithms and Computational Concerns......Page 362
Appendix A2. Orthogonality and Irrationality......Page 378
Appendix A3. An Interpolation Theorem......Page 388
Appendix A4. Inequalities for Generalized Polynomials in L p......Page 398
Appendix A5. Inequalities for Polynomials with Constraints......Page 423
Bibliography......Page 454
Notation......Page 473
Index......Page 479
Back Cover......Page 487
