Approximation by Max-Product Type Operators | springerprofessional.de

Springer Professional

nach oben

2016 | Buch

Kapitel lesen Erstes Kapitel lesen

Approximation by Max-Product Type Operators

verfasst von: Barnabás Bede, Lucian Coroianu, Sorin G. Gal

Verlag: Springer International Publishing

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

Einloggen, um Zugang zu erhalten

Über dieses Buch

This monograph presents a broad treatment of developments in an area of constructive approximation involving the so-called "max-product" type operators. The exposition highlights the max-product operators as those which allow one to obtain, in many cases, more valuable estimates than those obtained by classical approaches. The text considers a wide variety of operators which are studied for a number of interesting problems such as quantitative estimates, convergence, saturation results, localization, to name several.

Additionally, the book discusses the perfect analogies between the probabilistic approaches of the classical Bernstein type operators and of the classical convolution operators (non-periodic and periodic cases), and the possibilistic approaches of the max-product variants of these operators. These approaches allow for two natural interpretations of the max-product Bernstein type operators and convolution type operators: firstly, as possibilistic expectations of some fuzzy variables, and secondly, as bases for the Feller type scheme in terms of the possibilistic integral. These approaches also offer new proofs for the uniform convergence based on a Chebyshev type inequality in the theory of possibility.

Researchers in the fields of approximation of functions, signal theory, approximation of fuzzy numbers, image processing, and numerical analysis will find this book most beneficial. This book is also a good reference for graduates and postgraduates taking courses in approximation theory.

Anzeige

Inhaltsverzeichnis

Frontmatter

Chapter 1. Introduction and Preliminaries

Abstract

In this chapter we introduce the reader into the topic of the book and present some preliminaries useful for the next chapters.

Barnabás Bede, Lucian Coroianu, Sorin G. Gal

Chapter 2. Approximation by Max-Product Bernstein Operators

Section 2.1 of this chapter contains general results of approximation obtained by applying Theorem 1.1.2, Jackson-type estimates for some particular classes of functions and results of shape preserving.

Barnabás Bede, Lucian Coroianu, Sorin G. Gal

Chapter 3. Approximation by Max-Product Favard–Szász–Mirakjan Operators

Abstract

This chapter deals with the approximation and the shape preserving properties of the max-product Favard–Szász–Mirakjan operators, denoted by F _n ^(M)(f) in the non-truncated case, by \(\mathcal{T}_{n}^{(M)}(f)\) in the truncated case and attached to bounded functions f with only positive values. It is worth mentioning that this restriction can be dropped by attaching to bounded functions f of variable sign the new max-product type operators \(\overline{F}_{n}^{(M)}(f)(x) = F_{n}^{(M)}(f - a)(x) + a\), \(\overline{\mathcal{T}}_{n}^{(M)}(f)(x) = \mathcal{T}_{n}^{(M)}(f - a)(x) + a\), with a < inff. Indeed, by following the ideas in Theorem 2.9.1 (see also Subsection 1.1.3, Property C) it is easily seen that all the approximation and shape preserving properties proved for F _n ^(M)(f)(x) and \(\mathcal{T}_{n}^{(M)}(f)(x)\) below in this chapter remain valid for the max-product operators \(\overline{F}_{n}^{(M)}(f)(x)\) and \(\overline{\mathcal{T}}_{n}^{(M)}(f)(x)\).

Barnabás Bede, Lucian Coroianu, Sorin G. Gal

Chapter 4. Approximation by Max-Product Baskakov Operators

Abstract

This chapter studies the approximation and the shape preserving properties of the max-product Baskakov operators, denoted by V _n ^(M)(f) in the non-truncated case, by U _n ^(M)(f) in the truncated case and attached to bounded functions f with only positive values.

Barnabás Bede, Lucian Coroianu, Sorin G. Gal

Chapter 5. Approximation by Max-Product Bleimann–Butzer–Hahn Operators

Abstract

In this chapter the approximation and shape preserving properties of the max-product Bleimann–Butzer–Hahn operators attached to bounded positive functions are presented.

Barnabás Bede, Lucian Coroianu, Sorin G. Gal

Chapter 6. Approximation by Max-Product Meyer–König and Zeller Operators

Abstract

In this chapter the approximation and shape preserving properties of the max-product Meyer–König and Zeller operators, Z _n ^(M)(f)(x), are presented.

Barnabás Bede, Lucian Coroianu, Sorin G. Gal

Chapter 7. Approximation by Max-Product Interpolation Operators

Abstract

In this chapter we study the approximation properties of the following max-product operators of interpolation type: max-product Hermite–Fejér operator on Chebyshev knots of first kind, max-product Lagrange operator on Chebyshev knots of second kind, and max-product Lagrange operator on equidistant and on general Jacobi knots. An important characteristic of the approximation error estimates obtained is that they are all of Jackson-type, thus essentially improving those obtained in approximation by the counterpart linear interpolation operators.

Barnabás Bede, Lucian Coroianu, Sorin G. Gal

Chapter 8. Approximations by Max-Product Sampling Operators

Abstract

In this chapter we introduce and study the max-product sampling operators, which have applications to signal theory. Due to the fact that for bounded functions with positive values, the max-product sampling operators attached to them have nice properties, all the approximation results in this chapter are stated and proved under this restriction. But as it was already mentioned in Subsection 1.1.3, Property C, this restriction can easily be dropped by considering the construction used for the max-product Bernstein operator in Theorem 2.9.1. More precisely, if \(\mathcal{S}_{W,\varphi }^{(M)}\) is any max-product sampling operator defined in this chapter and \(f: \mathbb{R} \rightarrow \mathbb{R}\) is bounded and of variable sign, then it is easy to see that the new operators \(P_{W,\varphi }^{(M)}(f)(x) = \mathcal{S}_{W,\varphi }^{(M)}(f - a)(x) + a\), where \(a <\min \{ f(x);x \in \mathbb{R}\}\) and φ is the Fejér or the Whittaker kernel, keep all the approximation properties of the operator \(\mathcal{S}_{W,\varphi }^{(M)}\) (i.e., gives the same Jackson order of approximation, \(\omega _{1}(f;1/W)_{\mathbb{R}}\), keeps the interpolation properties and verifies the same saturation, local inverse, and localization results).

Barnabás Bede, Lucian Coroianu, Sorin G. Gal

Chapter 9. Global Smoothness Preservation Properties

Abstract

In this chapter we study the problem of partial global smoothness preservation in the cases of max-product Bernstein approximation operator, max-product Hermite–Féjer interpolation operator based on the Chebyshev nodes of first kind and max-product Lagrange interpolation operator based on the Chebyshev nodes of second kind.

Barnabás Bede, Lucian Coroianu, Sorin G. Gal

Chapter 10. Possibilistic Approaches of the Max-Product Type Operators

Abstract

It is known that the first proof of the uniform convergence for the Bernstein polynomials to a continuous function interprets them as a mean value of a random variable based on the Bernoulli distribution and uses the Chebyshev’s inequality in probability theory (see [33], or the more available [111]). The first main aim of this chapter is to give a proof for the convergence of the max-product Bernstein operators by using the possibility theory, which is a mathematical theory dealing with certain types of uncertainties and is considered as an alternative to probability theory. This new approach, which interprets the max-product Bernstein operator as a possibilistic expectation of a fuzzy variable having a possibilistic Bernoulli distribution, does not offer only a natural justification for the max-product Bernstein operators, but also allows to extend the method to other discrete max-product Bernstein type operators, like the max-product Meyer-König and Zeller operators, max-product Favard–Szász–Mirakjan operators, and max-product Baskakov operators.

Barnabás Bede, Lucian Coroianu, Sorin G. Gal

Chapter 11. Max-Product Weierstrass Type Functions

Abstract

Starting from the classical Weierstrass functions, in this chapter we introduce the so-called Weierstrass functions of max-product type, for which we prove that the set of the points of non-differentiability is uncountable, nowhere dense and of Lebesgue measure 0. Also, the fractal properties of these functions are studied.

Barnabás Bede, Lucian Coroianu, Sorin G. Gal

Backmatter

Titel: Approximation by Max-Product Type Operators
verfasst von: Barnabás Bede
Lucian Coroianu
Sorin G. Gal
Copyright-Jahr: 2016
Verlag: Springer International Publishing
Electronic ISBN: 978-3-319-34189-7
Print ISBN: 978-3-319-34188-0
DOI: https://doi.org/10.1007/978-3-319-34189-7

Premium Partner