Polynomials and Polynomial Inequalities

The most basic and important theorem concerning polynomials is the Fundamental Theorem of Algebra. This theorem, which tells us that every polynomial factors completely over the complex numbers, is the starting point for this book. Some of the intricate relationships between the location of the zeros of a polynomial and its coefficients are explored in Section 2. The equally intricate relationships between the zeros of a polynomial and the zeros of its derivative or integral are the subject of Section 1.3. This chapter serves as a general introduction to the body of theory known as the geometry of polynomials. Highlights of this chapter include the Fundamental Theorem of Algebra, the Eneström-Kakeya theorem, Lucas’ theorem, and Walsh’s two-circle theorem.

Chebyshev polynomials are introduced and their central role in problems in the uniform norm on [−1, 1] is explored. Sequences of orthogonal functions are then examined in some generality, although our primary interest is in orthogonal polynomials (and rational functions). The third section of this chapter is concerned with orthogonal polynomials; it introduces the most classical of these. These polynomials satisfy many extremal properties, similar to those of the Chebyshev polynomials, but with respect to (weighted) L ₂ norms. The final section of the chapter deals with polynomials with positive coefficients in various bases.

A Chebyshev space is a finite-dimensional subspace of C(A) of dimension n + 1 that has the property that any element that vanishes at n + 1 points vanishes identically. Such spaces, whose prototype is the space P _n of real algebraic polynomials of degree at most n, share with the polynomials many basic properties. The first section is an introduction to these Chebyshev spaces. A basis for a Chebyshev space is called a Chebyshev system. Two special families of Chebyshev systems, namely, Markov systems and Descartes systems, are examined in the second section. The third section examines the Chebyshev “polynomials” associated with Chebyshev spaces. These associated Chebyshev polynomials, which equioscillate like the usual Chebyshev polynomials, are extremal for various problems in the supremum norm. The fourth section studies particular Descartes systems on (0, ∞) in detail. These Systems, which we call Müntz systems, can be very explicitly orthonormalized on [0, 1], and this orthogonalization is also examined. The final section constructs Chebyshev “polynomials” associated with the Chebyshev spaces on [-1,1] and explores their various properties.

We give an extended treatment of when various Markov spaces are dense. In particular, we show that denseness, in many situations, is equivalent to denseness of the zeros of the associated Chebyshev polynomials. This is the principal theorem of the first section. Various versions of Weierstrass’ classical approximation theorem are then considered. The most important is in Section 4.2 where Müntz’s theorem concerning the denseness of span {1, x ^λ1, x ^λ2, …} is analyzed in detail. The third section concerns the equivalence of denseness of Markov spaces and the existence of unbounded Bernstein inequalities. In the final section we consider when rational functions derived from Markov systems are dense. Included is the surprising result that rational functions from a fixed infinite Müntz system are always dense.

The classical inequalities for algebraic and trigonometric polynomials are treated in the first section. These include the inequalities of Remez, Bernstein, Markov, and Schur. The second section deals with Markov’s and Bernstein’s inequalities for higher derivatives. The final section is concerned with the size of factors of polynomials.

Versions of Markov’s inequality for Müntz spaces, both in C[a,b] and L _p [0, 1], are given in the first section of this chapter. Bernstein- and Nikolskii-type inequalities are treated in the exercises, as are various other inequalities for Müntz polynomials and exponential sums. The second section provides inequalities, including most significantly a Remez-type inequality, for nondense Müntz spaces.

Precise Markov- and Bernstein-type inequalities are given for various classes of rational functions in the first section of this chapter. Extensions of the inequalities of Lax, Schur, and Russak are also presented, as are inequalities for self-reciprocal polynomials. The second section of the chapter is concerned with metric inequalities for polynomials and rational functions.