Analytic theory of polynomials

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Analytic theory of polynomials / Q.I. Rahman and G. Schmeisser

資料種別:

図書

出版情報:

Oxford : Clarendon Press, 2002

形態:

xiv, 742 p. ; 24 cm

シリーズ名:

London Mathematical Society monographs ; new ser., 26 <BA00134328>

著者名:

ISBN:

9780198534938 [0198534930]

書誌ID:

BA58795449

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書誌詳細

注記:

Includes bibliographical references (p. [681]-727) and indexes

言語:

英語

目次情報:

Introduction / 1：

The fundamental theorem of algebra / 1.1：

Symmetric polynomials / 1.2：

The continuity theorem / 1.3：

Orthogonal polynomials: general properties / 1.4：

The classical orthogonal polynomials / 1.5：

Harmonic and subharmonic functions / 1.6：

Tools from matrix analysis / 1.7：

Notes / 1.8：

Critical Points in Terms of Zeros / I：

Fundamental results on critical points / 2：

Convex hulls and the Gauss-Lucas theorem / 2.1：

Extensions of the Gauss-Lucas theorem / 2.2：

Average distances from a line or a point / 2.3：

Real polynomials and Jensen's theorem / 2.4：

Extensions of Jensen's theorem / 2.5：

More sophisticated methods / 2.6：

Circular domains and polar derivative / 3.1：

Laguerre's theorem, its variants, and applications / 3.2：

Apolarity / 3.3：

Grace's theorem and equivalent forms / 3.4：

More specific results on critical points / 3.5：

Products and quotients of polynomials / 4.1：

Derivatives of reciprocals of polynomials / 4.2：

Complex analogues of Rolle's theorem / 4.3：

Bounds for some of the critical points / 4.4：

Converse results / 4.5：

Applications to compositions of polynomials / 4.6：

Linear combination of rational functions / 5.1：

Complex analogues of the intermediate-value theorem / 5.2：

Linear combination of derivatives: Walsh's approach / 5.3：

Linear combination of derivatives: recursive approach / 5.4：

Multiplicative composition: Schur-Szego approach / 5.5：

Multiplicative composition: Laguerre's approach / 5.6：

Multipliers preserving the reality of zeros / 5.7：

Polynomials with real zeros / 5.8：

The span of a polynomial / 6.1：

Largest zero and largest critical point / 6.2：

Interlacing and the Hermite-Biehler theorem / 6.3：

Consecutive zeros and critical points / 6.4：

Refinement of Rolle's theorem / 6.5：

Conjectures and solutions / 6.6：

A conjecture of Popoviciu / 7.1：

A conjecture of Smale / 7.2：

The conjecture of Sendov / 7.3：

Zeros in Terms of Coefficients / 7.4：

Inclusion of all zeros / 8：

The Cauchy bound and its estimates / 8.1：

Various refinements / 8.2：

Multipliers and the Enestrom-Kakeya theorem / 8.3：

More general expansions / 8.4：

Orthogonal expansions with real coefficients / 8.5：

Alternative approach by matrix methods / 8.6：

Inclusion of some of the zeros / 8.7：

Inclusions in terms of a norm / 9.1：

Pellet's theorem and its consequences / 9.2：

Bounds in terms of some of the coefficients / 9.3：

The Landau-Montel problem / 9.4：

Number of zeros in an interval / 9.6：

The Budan-Fourier theorem and Descartes' rule / 10.1：

Exact count under a side condition / 10.2：

Extensions to pairs of conjugate zeros / 10.3：

Exact count by Sturm sequences / 10.4：

Exact count via quadratic forms / 10.6：

Number of zeros in a domain / 10.7：

General principles / 11.1：

Number of zeros in a sector / 11.2：

Number of zeros in a half-plane / 11.3：

The Routh--Hurwitz problem / 11.4：

Number of zeros in a disc / 11.5：

Distribution of zeros / 11.6：

Extremal Properties / 11.7：

Growth estimates / 12：

The Bernstein--Walsh lemma / 12.1：

The convolution method / 12.2：

The method of functionals / 12.3：

Local behaviour / 12.4：

Extensions to functions of exponential type / 12.6：

Mean values / 12.7：

Mean values on circles / 13.1：

A class of linear operators / 13.2：

Mean values on the unit interval / 13.3：

Derivative estimates on the unit disc / 13.4：

Bernstein's inequality and generalizations / 14.1：

Refinements / 14.2：

Conditions on the coefficients / 14.3：

Conditions on the zeros / 14.4：

Some special operators / 14.5：

Inequalities involving mean values / 14.6：

Derivative estimates on the unit interval / 14.7：

Inequalities of S. Bernstein and A. Markov / 15.1：

Extensions to higher-order derivatives / 15.2：

Two other extensions / 15.3：

Dependence of the bounds on the zeros / 15.4：

Some special classes / 15.5：

L[superscript p] analogues of Markov's inequality / 15.6：

Coefficient estimates / 15.7：

Polynomials on the unit circle / 16.1：

Coefficients of real trigonometric polynomials / 16.2：

Polynomials on the unit interval / 16.3：

References / 16.4：

List of notation

Index

Introduction / 1：

The fundamental theorem of algebra / 1.1：

Symmetric polynomials / 1.2：

The continuity theorem / 1.3：

Orthogonal polynomials: general properties / 1.4：

The classical orthogonal polynomials / 1.5：

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