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Preface

Polynomials pervade mathematics, and much that is beautiful in mathematics is related to polynomials. Virtually every branch of mathematics, from algebraic number theory and algebraic geometry to applied analysis, Fourier analysis, and computer science, has its corpus of theory arising from the study of polynomials. Historically, questions relating to polynomials, for example, the solution of polynomial equations, gave rise to some of the most important problems of the day. The subject is now much too large to attempt an encyclopedic coverage.

The body of material we choose to explore concerns primarily polynomials as they arise in analysis, and the techniques of the book are primarily analytic. While the connecting thread is the polynomial, this is an analysis book. The polynomials and rational functions we are concerned with are almost exclusively of a single variable.

We assume at most a senior undergraduate familiarity with real and complex analysis (indeed in most places much less is required). However, the material is often tersely presented, with much mathematics explored in the exercises, some of which are quite hard, many of which are supplied with copious hints, some with complete proofs. Well over half the material in the book is presented in the exercises. The reader is encouraged to at least browse through these. We have been much influenced by Pólya and Szegő’s classic “Problems and Theorems in Analysis” in our approach to the exercises. (Though unlike Pólya and Szegő we chose to incorporate the hints with the exercises.)

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The book is mostly self-contained. The text, without the exercises, provides an introduction to the material, but much of the richness is reserved for the exercises. We have attempted to highlight the parts of the theory and the techniques we find most attractive. So, for example, Müntz’s lovely characterization of when the span of a set of monomials is dense is explored in some detail. This result epitomizes the best of the subject: an attractive and nontrivial result with several attractive and nontrivial proofs.

There are excellent books on orthogonal polynomials, Chebyshev polynomials, Chebyshev systems, and the geometry of polynomials, to name but a few of the topics we cover, and it is not our intent to rewrite any of these. Of necessity and taste, some of this material is presented, and we have attempted to provide some access to these bodies of mathematics. Much of the material in the later chapters is recent and cannot be found in book form elsewhere.

Students who wish to study from this book are encouraged to sample widely from the exercises. This is definitely “hands on” material. There is too much material for a single semester graduate course, though such a course may be based on Sections 1.1 through 5.1, plus a selection from later sections and appendices. Most of the material after Section 5.1 may be read independently.

Not all objects labeled with “E” are exercises. Some are examples. Sometimes no question is asked because none is intended. Occasionally exercises include a statement like, “for a proof see … ”; this is usually an indication that the reader is not expected to provide a proof.

Some of the exercises are long because they present a body of material. Examples of this include E.11 of Section 2.1 on the transfinite diameter of a set and E.11 of Section 2.3 on the solvability of the moment problem. Some of the exercises are quite technical. Some of the technical exercises, like E.4 of Section 2.4, are included, in detail, because they present results that are hard to access elsewhere.

Acknowledgments

We would like to thank Dick Askey, Weiyu Chen, Carl de Boor, Karl Dilcher, Jens Happe, András Kroó, Doron Lubinsky, Gua-Hua Min, Paul Nevai, Allan Pinkus, József Szabados, Vilmos Totik, and Richard Varga. Their thoughtful and helpful comments made this a better book. We would also like to thank Judith Borwein, Maria Fe Elder, and Chiara Veronesi for their expert assistance with the preparation of the manuscript.

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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

Contents ix

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