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Polynomials and polynomial inequalities. (English) Zbl 0840.26002
New York, NY: Springer-Verlag. x, 480 p. (1995).
This attractive book deals with the analytic theory of polynomials and rational functions of one variable. An interesting feature of the book is the presentation of many of the results as series of exercises. This enables the book to cover much more ground than otherwise and enhances its value as a text for an advanced undergraduate or graduate course in approximation theory. The more difficult exerices are provided with liberal hints. Although some of the material in the early chapters would be found in many texts on approximation theory, the later chapters contain many results that have not previously appeared in textbooks. Most of these are results to which the authors have made major contributions. Thus the book would be of interest to professional mathematicians as well as to students.
The basic facts about approximation in the uniform norm are proved in the context of Chebyshev systems. This allows a uniform treatment of ordinary polynomials, trigonometric polynomials, spaces based on arbitrary powers \(\{x^{\lambda_k}\}\) (Müntz systems) and spaces spanned by \(\{1/(x- \lambda_k)\}\) and \(\{e^{\lambda_k x}\}\). The study of the Müntz systems is a recurring theme throughout the book. For example, in Chapter 3, we have the Müntz-Legendre polynomials. In Chapter 4, we have a presentation of the “Full Müntz theorem”, that if \(\{\lambda_k\}\) is a sequence of distinct positive numbers then the span of \(1,x^{\lambda_1}, x^{\lambda_2},\dots\) is dense in \(C[0, 1]\) if and only if \(\sum^\infty_{k= 1} \lambda_k/(\lambda^2_k+ 1)= \infty\). Notice that this allows 0 to be an accumulation point of \(\lambda_k\), a case not usually treated. A complete characterization is given for the density of the Müntz system in \(L_p[0, 1]\), for any \(p\geq 1\). The case of complex exponents is treated in an exercise. Müntz rationals are also considered in detail in this chapter. Chapter 5 is devoted to inequalities for polynomials. Of course, the basic Markov and Bernstein inequalities are considered, but there are also some recent results on inequalities for norms of factors of polynomials. Chapter 6 treats inequalities for Müntz spaces and contains, for example, a simplified proof of a sharp Markov-type inequality due to D. J. Newman. Chapter 7 is devoted to inequalities for certain spaces of rational functions and contains a proof of precise Markov-type and Bernstein-type inequalities for such functions.

MSC:
26-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to real functions
26C05 Real polynomials: analytic properties, etc.
26D05 Inequalities for trigonometric functions and polynomials
41A17 Inequalities in approximation (Bernstein, Jackson, Nikol’skiĭ-type inequalities)
30-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to functions of a complex variable
33-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to special functions
41-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to approximations and expansions
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