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Contents

PREFACE vii

CHAPTER 1 Introduction and Basic Properties 1

1.1 Polynomials and Rational Functions 1 1.2 The Fundamental Theorem of Algebra 11 1.3 Zeros of the Derivative 18

CHAPTER 2 Some Special Polynomials 29

2.1 Chebyshev Polynomials 29 2.2 Orthogonal Functions 41 2.3 Orthogonal Polynomials 57 2.4 Polynomials with Nonnegative Coefficients 79

CHAPTER 3 Chebyshev and Descartes Systems 91

3.1 Chebyshev Systems 92 3.2 Descartes Systems 100 3.3 Chebyshev Polynomials in Chebyshev Spaces 114 3.4 Müntz-Legendre Polynomials 125 3.5 Chebyshev Polynomials in Rational Spaces 139

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CHAPTER 4 Denseness Questions 154 4.1 Variations on the Weierstrass Theorem 154 4.2 Müntz’s Theorem 171 4.3 Unbounded Bernstein Inequalities 206 4.4 Müntz Rationals 218

CHAPTER 5 Basic Inequalities 227 5.1 Classical Polynomial Inequalities 227 5.2 Markov’s Inequality for Higher Derivatives 248 5.3 Inequalities for Norms of Factors 260

CHAPTER 6 Inequalities in Müntz Spaces 275 6.1 Inequalities in Müntz Spaces 275 6.2 Nondense Müntz Spaces 303

CHAPTER 7 Inequalities for Rational Function Spaces 320 7.1 Inequalities for Rational Function Spaces 320 7.2 Inequalities for Logarithmic Derivatives 344

APPENDIX A1 Algorithms and Computational Concerns 356 APPENDIX A2 Orthogonality and Irrationality 372 APPENDIX A3 An Interpolation Theorem 382 APPENDIX A4 Inequalities for Generalized Polynomials in L_{p} 392 APPENDIX A5 Inequalities for Polynomials with Constraints 417 BIBLIOGRAPHY 448 NOTATION 467 INDEX 473