[In PS Form]
Polynomials
and
Polynomial
Inequalities
[
Preface
| Contents
| Chapter [
1 |
2 |
3 |
4 |
5 |
6 |
7 ] |
Appendix [
1 |
2 |
3 |
4 |
5 ] |
Bibliography |
Notation |
Index ]
-
Preface
-
Polynomials pervade mathematics, and much that is
beautiful in mathematics is related to polynomials. Virtually every branch of
mathematics, from algebraic number theory and algebraic geometry to applied
analysis, Fourier analysis, and computer science, has its corpus of theory
arising from the study of polynomials. Historically, questions relating to
polynomials, for example, the solution of polynomial equations, gave
rise to some of the most important problems of the day.
The subject is now much too large to attempt an
encyclopedic coverage
-
Contents
-
Table of Contents
-
Chapter
1----
Introduction
and
Basics
Properties
-
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 Enestr\"om-Kakeya theorem, Lucas'
theorem, and Walsh's two-circle theorem.
-
Chapter
2----
Some
Special
Polynomials
-
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.
-
Chapter
3----
Chebyshev
and
Descartes
Systems
-
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 $$\left(x^{\lambda_0}, x^{\lambda_1}, \ldots \right)\,,
\qquad \lambda_0 < \lambda_1 < \ldots $$ on $(0,\infty)$ in detail. These systems,
which we call {\it M\"untz 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
\text{span}\left\{1\,,\, \frac{1}{x -a_1}\,,\, \ldots, \frac{1}{x -a_n}\right\}\,,
\qquad a_i \in {\Bbb R} \setminus [-1,1] on [-1,1] and explores their various properties.
-
Chapter
4----
Denseness
Questions
-
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 M\"untz's theorem concerning the denseness of
$\text{span} \{1, x^{\lambda_1}, x^{\lambda_2}, \ldots \}$ 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 M\"untz system are always dense.
-
Chapter
5----
Basic
Inequalities
-
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.
-
Chapter
6----
Inequalities
in
Muntz
Spaces
-
Versions of Markov's inequality for M\"untz 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 M\"untz
polynomials and exponential sums. The second section provides inequalities,
including most significantly a Remez-type inequality,
for nondense M\"untz spaces.
-
Chapter
7----
Inequalities
for
Rational
Function
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.
-
Appendix
1++++
Algorithms
and
Computational
Concerns
-
Appendix 1 presents some of the basic algorithms for computing with
polynomials and rational functions and discusses some of the complexity issues.
Included is a discussion of root finding methods. It requires very little background
and can essentially be read independently.
-
Appendix
2++++
Orthogonality
and
Irrationality
-
This appendix is an application of orthogonalization of particular M\"untz systems
to the proof of the irrationality of zeta(3) and some other familiar
numbers. It reproduces Ap\'ery's remarkable proof of the irrationality
of zeta(3) in the context of orthogonal systems
-
Appendix
3++++
An
Interpolation
Theorem
-
Appendix 3 presents an interpolation theorem for linear functions that is
used in Section 7.1. From this Haar's characterization of Chebyshev spaces
follows, as do alternate proofs of many of the basic inequalities.
-
Appendix
4++++
Inequalities
for
Generalized
Polynomials
in
Lp
-
Many inequalities for generalized polynomials are given
in this appendix. Of particular interest are the extensions of
virtually all the basic inequalities to L_p spaces. The principal tool
is a generalized version of Remez's inequality.
-
Appendix
5++++
Inequalities
for
Polynomials
With
Constraints
-
This appendix deals with inequalities for constrained polynomials.
Typically the constraints are on the location of the zeros, though various
coefficient constraints are also considered.
-
Bibliography
-
Notation
-
Definitions of the more commonly used spaces are given. The equation
numbers here are the same as the equation numbers in the text.
-
Index
-
Last Update : 17th September, 1995
Last Update : 21st September, 1995