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2001 | Buch

Orthogonal Polynomials for Exponential Weights

verfasst von: Eli Levin, Doron S. Lubinsky

Verlag: Springer New York

Buchreihe : CMS Books in Mathematics

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The analysis of orthogonal polynomials associated with general weights was a major theme in classical analysis in the twentieth century, and undoubtedly will continue to grow in importance in the future.
In this monograph, the authors investigate orthogonal polynomials for exponential weights defined on a finite or infinite interval. The interval should contain 0, but need not be symmetric about 0; likewise the weight need not be even. The authors establish bounds and asymptotics for orthonormal and extremal polynomials, and their associated Christoffel functions. They deduce bounds on zeros of extremal and orthogonal polynomials, and also establish Markov- Bernstein and Nikolskii inequalities.
The authors have collaborated actively since 1982 on various topics, and have published many joint papers, as well as a Memoir of the American Mathematical Society. The latter deals with a special case of the weights treated in this book. In many ways, this book is the culmination of 18 years of joint work on orthogonal polynomials, drawing inspiration from the works of many researchers in the very active field of orthogonal polynomials.

Inhaltsverzeichnis

Frontmatter
1. Introduction and Results
Abstract
Let I be a finite or infinite interval and let w: I → [0, ∞) be measurable with all power moments
$$ \int_{I} {{x^{n}}w(x)dx,{\text{ n = 0, 1, 2, 3,}}...} "$$
finite. Then we call w a weight and may define orthonormal polynomials
$$ {p_{n}}(x) = {p_{n}}(w,{\text{ }}x) = {\gamma _{n}}(w){x^{n}} + \cdot \cdot \cdot ,{\gamma _{n}}(w) > 0, "$$
satisfying
$$ \int_{I} {{p_{n}}{p_{m}}w = {d_{{mn}}},m, n = 0, 1, 2,... .} "$$
Eli Levin, Doron S. Lubinsky
2. Weighted Potential Theory: The Basics
Abstract
We present some of the basics of weighted potential theory, and derive useful identities involving a t .
Eli Levin, Doron S. Lubinsky
3. Basic Estimates for Q, a t
Abstract
In this chapter, we establish some basic estimates for Q, a t , and similar quantities. We shall deal with various classes of weights on I=(c, d). Recall that I may be finite or infinite and contains 0 as an interior point. We shall often state results only for the interval (0, d); the reader will easily observe what is the analogous statement for (c, 0).
Eli Levin, Doron S. Lubinsky
4. Restricted Range Inequalities
Abstract
We have already seen that for Q convex, and not identically vanishing polynomials P of degree ≦ n, there holds the Mhaskar-Saff inequality:
$$ \left| {P{e^{{ - Q}}}} \right|(x) < {\left\| {PW} \right\|_{{{L_{\infty }}({\Delta _{n}})}}},x \in I\backslash {\Delta _{n}}. "$$
Eli Levin, Doron S. Lubinsky
5. Estimates for Measure and Potential
Abstract
In this chapter, we obtain upper and lower bounds for the equilibrium density σ t (x) and for the associated potential V μt . The lower bounds are easy to obtain, and minimal assumptions on W are needed:
Eli Levin, Doron S. Lubinsky
6. Smoothness of σ t
Abstract
The smoothness of σ t plays a role in discretising the potential to obtain weighted polynomial approximations. In this chapter, we establish various levels of smoothness of σ t under corresponding conditions on Q.
Eli Levin, Doron S. Lubinsky
7. Weighted Polynomial Approximation
Abstract
In this chapter, we establish the existence of weighted polynomial approximations that are a prerequisite to the estimates and asymptotics in subsequent chapters. We search for polynomials P n of degree n such that P n W approximates 1 in some sense on [a −n, a n ]. There are several approaches to this problem ([48], [86], [115], [136], [66], [178]), most based on discretisation of the potential. The most successful one has been given by Totik [178], and this is the method that we employ. Because of our need to approximate almost up to a ±n , our results do not follow from those in [178], [180], [182].
Eli Levin, Doron S. Lubinsky
8. Asymptotics of Extremal Errors
Abstract
In this chapter, we establish asymptotics of the extremal errors
$$ {E_{{n,p}}}(W): = \mathop{{\inf }}\limits_{{P(x) = {x^{n}} + \cdot \cdot \cdot }} \left\| {PW} \right\|{L_{p}}(I), "$$
where the inf is taken over all monic polynomials P of degree n. In order to state our result in compact form, we need some notation. For a non-negative h : [−1,1] → ℝ, let
$$ G\left[ h \right]: = \exp \left( {\frac{1}{\pi }\int_{{ - 1}}^{1} {\frac{{\log h(u)}}{{\sqrt {{1 - {u^{2}}}} }}du} } \right) "$$
denote the geometric mean of h. Recall also that
$$ W_{n}^{{_{*}}}\left( u \right): = W\left( {L_{n}^{{\left[ { - 1} \right]}}\left( u \right)} \right),{\text{ u}} \in {\text{Ln}}\left( {\text{I}} \right){\text{,}} "$$
where L n is the linear map of [a −n, an] onto [−1,1] and L n [−1] is its inverse. Finally, let
$$ {\kappa _{p}}: = \left\{ {\begin{array}{*{20}{c}} {{{\left( {\sqrt {\pi } \Gamma \left( {\frac{{p + 1}}{2}} \right)/\Gamma \left( {\frac{p}{2} + 1} \right)} \right)}^{{1/p}}}} \hfill \\ {1,} \hfill \\ \end{array} } \right.\begin{array}{*{20}{c}} {p < \infty } \\ {p = \infty } \\ \end{array} . "$$
Eli Levin, Doron S. Lubinsky
9. Christoffel Functions
Abstract
Christoffel functions are crucially important in analysis of orthogonal poly-nomials and weighted approximation theory [146]. In this chapter we shall estimate generalized and classical L p Christoffel functions for 0 < p≤∞ using the polynomials constructed in Chapter 7. We shall also establish asymptotics for classical Christoffel functions.
Eli Levin, Doron S. Lubinsky
10. Markov-Bernstein and Nikolskii Inequalities
Abstract
In this chapter, we shall prove Markov-Bernstein inequalities and Nikolskii inequalities. We begin with the former. We shall make substantial use of the function ϕ t defined by (9.18) and (9.19).
Eli Levin, Doron S. Lubinsky
11. Zeros of Orthogonal Polynomials
Abstract
Recall that given a weight W 2 on the finite or infinite interval I = (c, d), its nth orthonormal polynomial p n (W 2,x) has zeros \( \left\{ {{x_{{jn}}}} \right\}\begin{array}{*{20}{c}} n \hfill \\ {j = 1} \hfill \\ \end{array} , \), where
$$ c < {x_{{nn}}} < {x_{{n - 1,n}}} < \cdot \cdot \cdot < {x_{{2n}}}{ < _{{1n}}} < d. "$$
Eli Levin, Doron S. Lubinsky
12. Bounds on Orthogonal Polynomials
Abstract
Perhaps the most significant result of this work is the following uniform bound on the orthogonal polynomials throughout the interval of orthogonality:
Eli Levin, Doron S. Lubinsky
13. Further Bounds and Applications
Abstract
In this chapter, we obtain further upper (and lower) bounds on orthogonal polynomials and on their L p norms. We also estimate fundamental polynomials of Lagrange interpolation, and spacing of zeros of orthogonal polynomials. We shall often need more than WF(lip1/2). Recall from Chapter 1 that we defined WF(lip1/2+) if both WF(lip1/2) and for each L>1, there exists C>0 and t 0 such that
$$ Q'\left( {{a_{{Lt}}}} \right)/Q'\left( {{a_{t}}} \right) \geqslant 1 + C,\left| t \right| \geqslant {t_{0}}. "$$
Eli Levin, Doron S. Lubinsky
14. Asymptotics of Extremal Polynomials
Abstract
In this chapter, we establish mean asymptotics on the real line for extremal polynomials, and also their asymptotics in the plane. Our approach follows very closely that in [101] and also [114].
Eli Levin, Doron S. Lubinsky
15. Asymptotics of Orthonormal Polynomials
Abstract
In this chapter, we establish pointwise asymptotics of the orthonormal polynomials p n (W 2, x) for x in the interval of orthogonality, as well as asymptotics for the associated recurrence coefficients. We shall also reformulate some of the results of the previous chapters for this special case.
Eli Levin, Doron S. Lubinsky
Backmatter
Metadaten
Titel
Orthogonal Polynomials for Exponential Weights
verfasst von
Eli Levin
Doron S. Lubinsky
Copyright-Jahr
2001
Verlag
Springer New York
Electronic ISBN
978-1-4613-0201-8
Print ISBN
978-1-4612-6563-4
DOI
https://doi.org/10.1007/978-1-4613-0201-8