nach oben

2004 | Buch

Kapitel lesen Erstes Kapitel lesen

Approximation Theory Using Positive Linear Operators

verfasst von: Radu Păltănea

Verlag: Birkhäuser Boston

Enthalten in: Professional Book Archive

Einloggen, um Zugang zu erhalten

Über dieses Buch

We deal in this work with quantitative results in the pointwise approximation of func tions by positive linear functionals and operators. One of the main objectives is to obtain estimates for the degree of approximation in terms of various types of second order moduli of continuity. In the category of sec ond order moduli we include both classical and newly introduced moduli. Particular attention is paid to optimizing the constants appearing in such estimates. In the last decades, the study of linear positive operators with the aid of second order moduli was intensive, thanks to their refinements in characterization of the smoothness of functions. As promoters of this direction of research we mention Yu. Brudnyi, G. Freud, and J. Petree. Our approach is more akin to the approach taken by H. Gonska, who obtained the first general estimates for second order moduli with precise constants and with free parameters. Two new methods will be presented. The first one, based on decomposition of functionals and the use of moments, can be applied to diverse types of moduli and leads to simple estimates. The second method gives sufficient conditions for obtaining absolute optimal constants. The benefits of these more direct methods, compared with the known method based on K-functionals, consist in the improvement and even the optimization of the constants, and in the generalization of the framework.

Inhaltsverzeichnis

Frontmatter

1. Introduction

Abstract

The most constructive proofs of the Weierstrass theorem concerning the approximation of continuous functions on a compact interval by polynomials use some sequences of linear positive operators. So do the classical operators of Gauss—Weierstrass, Landau, Vallée-Poussin, Jackson, and Bernstein. We begin by constructing and studying a large class of such sequences of approximation operators.

Radu Păltănea

2. Estimates with Second Order Moduli

Abstract

In this chapter we continue the study of estimating the degree of an approximation using general linear positive operators by considering combinations of first and second order moduli, in terms of the moments of order 0, 1, and 2, see Remark 1.2.4. Estimates with such combinations of first and second order modulus, (and also with the absolute value of the function, which can be regarded as a modulus of order 0) are more refined then estimates using only the first modulus. A first observation is that, from estimates with the second order modules, one can derive estimates with the first order modulus. A second observation is the fact that such combinations decompose the error of approximation in three components, corresponding to three specific features of the functions that affect the error: amplitude, deviation from the linear functions, and deviation from the polynomials of degree 2. Roughly speaking, these moduli measure the deviation from the test functions of the algebraic Chebychev system.

Radu Păltănea

3. Absolute Optimal Constants

Abstract

We point out the optimality of the estimate given in Theorem 2.2.2, for s = 2, in a stronger sense than the optimality of the constants. Let x be an interior point of I and denote by U_x (I) the family of all linear positive functionals F defined on a subspace V_F ⊂ℱ_b(I), with the property F(e₀) = 1, F(e₁) = x and which are admissible related to the point x.

Radu Păltănea

4. Estimates for the Bernstein Operators

Abstract

The Bernstein operators B_n, n ∈ ℕ assign to each function F ∈ ℱ[0, 1], the polynomials

$$ {{B}_{n}}(f,x): = \sum\limits_{{k = 0}}^{n} {{{p}_{{n,k}}}(x) \cdot f\left( {\frac{k}{n}} \right),x \in [0,1],\;where} \;{{p}_{{n,k}}}(x): = \left( {\begin{array}{*{20}{c}} n \\ k \\ \end{array} } \right){{x}^{k}}{{(1 - x)}^{{n - k}}}. $$

(4.1)

Radu Păltănea

5. Two Classes of Bernstein Type Operators

Abstract

One of the most natural extensions of the Bernstein operators was made by H. Brass [17]. These operators are of the form

$$ {{P}_{n}}(f,x): = \sum\limits_{{k = 0}}^{n} {f\left( {\frac{k}{n}} \right){{q}_{{n,k}}}(x),f \in F[0,1],x \in [0,1],n \in \mathbb{N},} $$

(5.1)

where q_n,k are polynomials of degree n that are positive on the interval [0, 1] and are such that the following properties are true:

P_n is a linear positive operator

P_n preserves linear functions

P_n preserves the degree of any polynomial of degree at most n and

P_n preserves the convexity of higher order k, for any k ≥ -1, (see Definition 1.1.1).

Radu Păltănea

6. Approximation Operators for Vector-Valued Functions

Abstract

Let (E, < ·, · >) be a Euclidean space with the norm denoted by ‖ · ‖. Let I = [a, b], a < b be a real interval. Let C(I, E) be the space of continuous functions, endowed with the sup-norm denoted by ‖ · ‖_I and denote by C^k(I, E), k ≥ 1, the subspace of functions having a continuous derivative of order k on I. If φ : I → ℝ and w ∈ E we denote by φw, the function (φw)(x) = φ(x)w.

Radu Păltănea

Backmatter

Titel: Approximation Theory Using Positive Linear Operators
verfasst von: Radu Păltănea
Verlag: Birkhäuser Boston
Electronic ISBN: 978-1-4612-2058-9
Print ISBN: 978-0-8176-4350-8
DOI: https://doi.org/10.1007/978-1-4612-2058-9

Springer Professional

Über dieses Buch

Inhaltsverzeichnis

Frontmatter

1. Introduction

2. Estimates with Second Order Moduli

3. Absolute Optimal Constants

4. Estimates for the Bernstein Operators

5. Two Classes of Bernstein Type Operators

6. Approximation Operators for Vector-Valued Functions

Backmatter