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2017 | Buch

Preliminaries Kapitel lesen Erstes Kapitel lesen

Bernstein Operators and Their Properties

verfasst von: Jorge Bustamante

Verlag: Springer International Publishing

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

This book provides comprehensive information on the main aspects of Bernstein operators, based on the literature to date. Bernstein operators have a long-standing history and many papers have been written on them. Among all types of positive linear operators, they occupy a unique position because of their elegance and notable approximation properties.

This book presents carefully selected material from the vast body of literature on this topic. In addition, it highlights new material, including several results (with proofs) appearing in a book for the first time. To facilitate comprehension, exercises are included at the end of each chapter.

The book is largely self-contained and the methods in the proofs are kept as straightforward as possible. Further, it requires only a basic grasp of analysis, making it a valuable and appealing resource for advanced graduate students and researchers alike.

Inhaltsverzeichnis

Frontmatter
Chapter 1. Preliminaries
Abstract
Throughout the book we will use the notation \(\varphi (x) = \sqrt{x(1 - x)},\quad x \in [0,1]\) and, for a non negative integer j, \(e_{j}(x) = x^{j},\quad x \in \mathbb{R}.\) Moreover, \(\mathbb{P}_{n}\) denotes the family of all algebraic polynomials of degree no greater than n.
Jorge Bustamante
Chapter 2. Basic Properties of Bernstein Operators
Abstract
At the beginning of the twentieth century several proofs of the convergence of Bernstein polynomials were published. In particular, we note the ones due to Chlodovski [75], Sierpinski [345], Wundheiler [411], Kac ([210] and [211]), and Kendall [217]. The first estimates in terms of a modulus of continuity were established by Popoviciu [312] in 1935 (see also [288, p. 197]).
Jorge Bustamante
Chapter 3. Bernstein Polynomials as Linear Operators
Abstract
Recall that, if (X, ∥ ∘ ∥  X ) and (Y, ∥ ∘ ∥  Y ) are normed spaces and L: X → Y is a continuous linear operator, then the norm of L is defined as \(\Vert L\Vert _{X\rightarrow Y } =\sup \{\Vert L(x)\Vert _{Y }\,:\,\Vert x\Vert _{X} \leq 1\}.\) Moreover, a linear operator L: X → Y is continuous if and only if ∥ L ∥  X → Y  < ∞.
Jorge Bustamante
Chapter 4. Upper Error Estimates of Bernstein Operators

In the first section of this chapter we present some auxiliary results. Concretely, we give some properties concerning positive linear operators.

Jorge Bustamante
Chapter 5. Bernstein-Type Inequalities
Abstract
For studying inverse results it is important to have on hand some particular expressions for the derivatives of Bernstein operators, as well as various estimates involving these derivatives. We first study pointwise estimates. As before, we set \(\varphi (x) = \sqrt{x(1 - x)}\).
Jorge Bustamante
Chapter 6. Converse Results
Abstract
How much are the smoothness of a function and its rate of approximation by Bernstein operators related? What can be said about the structural properties of a function f ∈ C[0, 1], if some information on the behavior of the sequence { ∥ f − B n ( f) ∥ } is known? How do the singularities of a function influence the order of its approximation by Bernstein polynomials? There are several expressions concerning the connection between the rate of convergence of Bernstein polynomials and the smoothness of the function been approximated. In a previous chapter we have estimated the error in terms of different moduli of smoothness. By a converse result we mean an estimate of the moduli of smoothness in terms of the error of approximation. Converse results are much more difficult to prove than direct ones.
Jorge Bustamante
Chapter 7. Bernstein Operators and a Special Class of Functions
Abstract
In this section we present the most important limitations of the Bernstein polynomials.
Jorge Bustamante
Chapter 8. Iterates of Bernstein Polynomials
Abstract
For f ∈ C[0, 1] and \(n,j \in \mathbb{N}\), one has
$$\displaystyle{B_{n}^{j}(\,f,x) =\sum \limits _{ k=0}^{n}(\lambda _{ n,k})\,^{j}P_{ n,k}^{{\ast}}\mu _{ k}^{(n)}(\,f).}$$
Jorge Bustamante
Chapter 9. Linear Combinations of Bernstein Polynomials
Abstract
For complex functions we do not find such a good variety of results as the ones presented in previous chapters. For instance, there are no converse results.
Jorge Bustamante
Chapter 10. Final Comments
Abstract
For complex functions we do not find such a good variety of results as the ones presented in previous chapters. For instance, there are no converse results.
Jorge Bustamante
Backmatter
Metadaten
Titel
Bernstein Operators and Their Properties
verfasst von
Jorge Bustamante
Copyright-Jahr
2017
Electronic ISBN
978-3-319-55402-0
Print ISBN
978-3-319-55401-3
DOI
https://doi.org/10.1007/978-3-319-55402-0

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