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2003 | Buch

Basic Principles and Facts Kapitel lesen Erstes Kapitel lesen

Multivariate Polynomial Approximation

verfasst von: Manfred Reimer

Verlag: Birkhäuser Basel

Buchreihe : ISNM International Series of Numerical Mathematics

Enthalten in: Professional Book Archive

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SUCHEN

Über dieses Buch

Multivariate polynomials are a main tool in approximation. The book begins with an introduction to the general theory by presenting the most important facts on multivariate interpolation, quadrature, orthogonal projections and their summation, all treated under a constructive view, and embedded in the theory of positive linear operators. On this background, the book gives the first comprehensive introduction to the recently developped theory of generalized hyperinterpolation. As an application, the book gives a quick introduction to tomography. Several parts of the book are based on rotation principles, which are presented in the beginning of the book, together with all other basic facts needed.

Inhaltsverzeichnis

Frontmatter

Introduction

Frontmatter
Chapter 1. Basic Principles and Facts
Abstract
We investigate polynomial approximations to functions
$$F:D \to mathbb{R}$$
(1.1)
where D is a nonempty compact subset of ℝ r ,γ∈ℕ preferably in the uniform norm, but occasionally also in the quadratic average norm. The function is called multivariate, if r ≥ 2. C(D) denotes the space of all continuous functions (1.1) which is provided with the norm
$$ {\left\| F \right\|_\infty }: = {\left\| F \right\|_D}: = \max \left\{ {\left| {F\left( x \right)} \right|:x \in D} \right\}.$$
.
Manfred Reimer
Chapter 2. Gegenbauer Polynomials
Abstract
In the constructive theory of spherical functions the Gegenbauer polynomials play an important role. Apart from constant factors they are certain Jacobi polynomials. For α, β> −1, the indices, the Jacobi polynomials P µ (α,β) of degree μ∈ℕ0are defined as orthogonal polynomials with respect to the inner product
$$ \left( {f,g} \right): = \int\limits_{ - 1}^1 {f\left( x \right)g\left( x \right){{\left( {1 - x} \right)}^\alpha }{{\left( {1 + x} \right)}^\beta }dx},$$
(2.1)
α, β> -1, which are normalized by the condition
$$ P_\mu ^{\alpha,\beta }\left( 1 \right) = \left( {\begin{array}{*{20}{c}} {\mu + \alpha } \\ \mu \end{array}} \right).$$
(2.2)
.
Manfred Reimer

Approximation Means

Frontmatter
Chapter 3. Multivariate Polynomials
Abstract
The theory of multivariate polynomial approximation is characterized by a great variety of polynomials which can be used, and also by a great richness of geometric situations which occur. This chapter presents the most important facts on multivariate polynomials.
Manfred Reimer
Chapter 4. Polynomials on Sphere and Ball
Abstract
In this chapter we investigate the space ℙ r (S r-1) of polynomial restrictions onto the unit sphere S r-1, and its subspaces. The elements of ℙ r (S r-1) are called spherical polynomials, the elements of \(\mathop {{\Bbb H}_\mu ^r}\limits^* \) spherical harmonics of degree μ.
Manfred Reimer

Multivariate Approximation

Frontmatter
Chapter 5. Approximation Methods
Abstract
We investigate polynomial approximations to multivariate functions which are defined by linear operators. The corresponding theory is ruled by some important principles and theorems, which we present in the beginning.
Manfred Reimer
Chapter 6. Approximation on the Sphere
Abstract
This chapter is devoted to the particular, but important case where the domain is the unit sphere S r-1, where γ ∈ ℕ\{1}. This is the simplest rotation-invariant domain, so the theory is basic, and taking a plain and valid form. Several results can be transferred to the ball, see Section 7, with an important application to tomography in Section 8.
Manfred Reimer
Chapter 7. Approximation on the Ball
Abstract
In this chapter the domain is the unit ball B r , r ≥ 2. Of course, all that has been said in Section 5 with respect to an arbitrary compact domain is valid also for B r . But there are particulars now which require a separate consideration.
Manfred Reimer

Applications

Frontmatter
Chapter 8. Tomography
Abstract
In this section we consider a recovery problem for real functions F, which are hidden in a given function space X and which are to be reconstructed from the values λF, where λ varies in a family Λ of linear functionals on X. In practice, F will be some density function, while the values λF are accessible to measurement.
Manfred Reimer
Backmatter
Metadaten
Titel
Multivariate Polynomial Approximation
verfasst von
Manfred Reimer
Copyright-Jahr
2003
Verlag
Birkhäuser Basel
Electronic ISBN
978-3-0348-8095-4
Print ISBN
978-3-0348-9436-4
DOI
https://doi.org/10.1007/978-3-0348-8095-4