Analytic Theory of Polynomials

Transcription

1 Analytic Theory of Polynomials Q. I. Rahman Universite de Montreal and G. Schmeisser Universitat Erlangen -Niirnberg CLARENDON PRESS-OXFORD 2002

2 1 Introduction The fundamental theorem of algebra Symmetric polynomials The continuity theorem Orthogonal polynomials: general properties The classical orthogonal polynomials Harmonic and subharmonic functions Tools from matrix analysis Notes 61 I CRITICAL POINTS IN TERMS OF ZEROS Fundamental results on critical points Convex hulls and the Gauss-Lucas theorem Extensions of the Gauss-Lucas theorem Average distances from a line or a point Real polynomials and Jensen's theorem Extensions of Jensen's theorem Notes 91 More sophisticated methods Circular domains and polar derivative Laguerre's theorem, its variants, and applications Apolarity Grace's theorem and equivalent forms Notes 114 More specific results on critical points Products and quotients of polynomials Derivatives of reciprocals of polynomials Complex analogues of Rolle's theorem Bounds for some of the critical points Converse results Notes ' 137 Applications to compositions of polynomials Linear combination of rational functions Complex analogues of the intermediate-value theorem Linear combination of derivatives: Walsh's approach 148

3 xii 5.4 Linear combination of derivatives: recursive approach Multiplicative composition: Schur-Szego approach Multiplicative composition: Laguerre's approach Multipliers preserving the reality of zeros Notes Polynomials with real zeros The span of a polynomial Largest zero and largest critical point Interlacing and the Hermite-Biehler theorem Consecutive zeros and critical points Refinement of Rolle's theorem Notes Conjectures and solutions A conjecture of Popoviciu A conjecture of Smale The conjecture of Sendov Notes 237 II ZEROS IN TERMS OF COEFFICIENTS 8 Inclusion of all zeros The Cauchy bound and its estimates Various refinements Multipliers and the Enestrom-Kakeya theorem More general expansions Orthogonal expansions with real coefficients Alternative approach by matrix methods Notes Inclusion of some of the zeros Inclusions in terms of a norm Pellet's theorem and its consequences Bounds in terms of some of the coefficients Orthogonal expansions with real coefficients The Landau-Montel problem Notes Number of zeros in an interval The Budan-Fourier theorem and Descartes' rule Exact count under a side condition Extensions to pairs of conjugate zeros More general expansions Exact count by Sturm sequences Exact count via quadratic forms Notes 350

4 xiii 11 Number of zeros in a domain General principles Number of zeros in a sector Number of zeros in a half-plane The Routh-Hurwitz problem Number of zeros in a disc Distribution of zeros Notes 392 III EXTREMAL PROPERTIES 12 Growth estimates The Bernstein-Walsh lemma The convolution method The method of functionals Various refinements Local behaviour Extensions to functions of exponential type Notes Mean values Mean values on circles A class of linear operators Mean values on the unit interval Notes Derivative estimates on the unit disc Bernstein's inequality and generalizations Refinements Conditions on the coefficients Conditions on the zeros Some special operators Inequalities involving mean values Notes Derivative estimates on the unit interval Inequalities of S. Bernstein and A. Markov Extensions to higher-order derivatives Two other extensions Dependence of the bounds on the zeros Some special classes L p analogues of Markov's inequality Notes 622

5 xiv 16 Coefficient estimates Polynomials on the unit circle Coefficients of real trigonometric polynomials Polynomials on the unit interval Notes 677 References 681 List of notation 729 Index 733