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Topics in polynomials: extremal problems, inequalities, zeros. (English) Zbl 0848.26001
Singapore: World Scientific. xiii, 821 p. (1994).

There is no doubt that this is a very useful work compiling enormous researches carried out on the subject by many mathematical scientists including each of the three authors, especially by Dragoslav S. Mitrinović (1908-1995). It presents a systematic survey (with interesting historical notes) of some of the most important (classical as well as recent) results on the analysis of polynomials and their derivatives. Thus, not only does the book include all the relevant fundamental results on the subject with their proofs, it also provides a rather detailed account of the most recent developments concerning extremal properties of polynomials and their derivatives, and properties and characteristics of their zeros.

This 821-page work comprises a total of 7 chapters, which collectively cite some 1200 references, a Symbol Index of 3 pages, a Name Index of 16 pages, and a Subject Index of 19 pages. A brief account of the chapter-wise content of this book is being given below.

Chapter 1 presents a review of the classical results on algebraic polynomials in one and several variables, as well as on trigonometric polynomials. The Fejér-Riesz representation of nonnegative trigonometric polynomials, the Lorentz representation of polynomials, basic properties of orthogonal polynomials, the classical orthogonal polynomials, and other polynomial systems (such as the Appell polynomial systems) are discussed here.

Chapter 2 considers polynomial inequalities involving algebraic and trigonometric polynomials. Inequalities satisfied by the zeros, by the moments, by the coefficients, by derivatives, and so on, are presented here.

Chapter 3 discusses the distribution of zeros of algebraic polynomials, including such classical results as the Gauss-Lucas theorem, the Sendov-Ilieff conjecture and related topics, and bounds for the zeros and for their number in a given domain, including the Eneström-Kakeya theorem and its generalizations.

Chapter 4 presents inequalities associated with trigonometric sums. Many classical results (such as the inequalities of Fejér-Gronwall-Jackson, Young, Rogosinski, and Szegö) are included here. The authors also consider various positivity and monotonicity results in this chapter.

Chapter 5 discusses various extremal problems for polynomials. The topics considered here include polynomials with maximal norm and estimates for coefficients, incomplete polynomials and weighted norm inequalities, and inequalities of Nikol’skij type.

Chapter 6 deals with extremal problems of Markov and Bernstein type. Here, the authors begin with inequalities of the Markov and Bernstein type, and then discuss extremal problems for restricted polynomial classes and extremal problems in a circle.

Finally, in Chapter 7, the authors present various applications of polynomials. The topics considered in this chapter include least squares approximation with constraints, simultaneous approximation, the Bernstein conjecture in approximation theory, and applications in computer and geometric design.

Each chapter is concluded with a set of references cited in the chapter. Indeed the authors have succeeded in their attempt to present the material in an integrated and self-contained fashion.

This is a well-written book on a widely useful topic. It is strongly recommended not only to the mathematical specialist, but also to all those researchers in the applied and computational sciences who make frequent use of polynomials as a tool. Of course, libraries will also benefit greatly by including this book in their cherished collection.

26-02Research monographs (real functions)
26C05Polynomials: analytic properties (real variables)
26C10Polynomials: location of zeros (real variables)
26D05Inequalities for trigonometric functions and polynomials
30-02Research monographs (functions of one complex variable)
12-02Research monographs (field theory)
41-02Research monographs (approximations and expansions)
42-02Research monographs (Fourier analysis)
65-02Research monographs (numerical analysis)
41A17Inequalities in approximation (Bernstein, Jackson, Nikol’skiĭ-type inequalities)
42A05Trigonometric polynomials, inequalities, extremal problems