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Cover: 9780817643508 | Approximation Theory Using Positive Linear Operators | Radu Paltanea
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Approximation Theory Using Positive Linear Operators
Taschenbuch von Radu Paltanea
Sprache: Englisch

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Beschreibung
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.
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.
Zusammenfassung

This work treats quantitative aspects of the approximation of functions using positive linear operators. The theory of these operators has been an important area of research in the last few decades, particularly as it affects computer-aided geometric design. New and efficient methods, applicable to Bernstein operators and to diverse concrete moduli, are presented in this book. The advantages of these methods consist in obtaining improved and even optimal estimates, as well as in broadening the applicability of the results.

This monograph will be of interest to those working in the field of approximation or functional analysis. Requiring only familiarity with the basics of approximation theory, the book may serve as a good supplementary text for courses in approximation theory, or as a reference text on the subject.

Inhaltsverzeichnis
1 Introduction.- 1.1 Operators and functionals. Moduli of continuity.- 1.2 Approximation of functions by sequences of positive linear operators.- 2 Estimates with Second Order Moduli.- 2.1 A general approach.- 2.2 Estimates with moduli ?2? and ?2?.- 2.3 Estimates with modulus ?2d.- 2.4 Estimates with modulus ?2dd.- 2.5 Estimates with Ditzian-Totik modulus.- 3 Absolute Optimal Constants.- 3.1 Introduction.- 3.2 Discrete functionals and the classical second order modulus ?2.- 3.3 General functionals and the second order modulus with parameter ?2?.- 4 Estimates for the Bernstein Operators.- 4.1 Various types of estimates.- 4.2 Best constant in the estimate with modulus ?2.- 4.3 Global smoothness preservation.- 5 Two Classes of Bernstein Type Operators.- 5.1 Generalized Brass type operators.- 5.2 Generalized Durrmeyer type operators.- 6 Approximation Operators for Vector-Valued Functions.- 6.1 Approximation of functions with real argument.- 6.2 Approximation of functions with vector argument.- References.
Details
Erscheinungsjahr: 2004
Fachbereich: Analysis
Genre: Mathematik
Rubrik: Naturwissenschaften & Technik
Medium: Taschenbuch
Inhalt: x
202 S.
ISBN-13: 9780817643508
ISBN-10: 0817643508
Sprache: Englisch
Herstellernummer: 10996522
Ausstattung / Beilage: Paperback
Einband: Kartoniert / Broschiert
Autor: Paltanea, Radu
Herausgeber: George A Anastassiou
Hersteller: Birkh„user Boston
Birkhäuser Boston
Maße: 235 x 155 x 12 mm
Von/Mit: Radu Paltanea
Erscheinungsdatum: 17.09.2004
Gewicht: 0,335 kg
Artikel-ID: 102423511
Zusammenfassung

This work treats quantitative aspects of the approximation of functions using positive linear operators. The theory of these operators has been an important area of research in the last few decades, particularly as it affects computer-aided geometric design. New and efficient methods, applicable to Bernstein operators and to diverse concrete moduli, are presented in this book. The advantages of these methods consist in obtaining improved and even optimal estimates, as well as in broadening the applicability of the results.

This monograph will be of interest to those working in the field of approximation or functional analysis. Requiring only familiarity with the basics of approximation theory, the book may serve as a good supplementary text for courses in approximation theory, or as a reference text on the subject.

Inhaltsverzeichnis
1 Introduction.- 1.1 Operators and functionals. Moduli of continuity.- 1.2 Approximation of functions by sequences of positive linear operators.- 2 Estimates with Second Order Moduli.- 2.1 A general approach.- 2.2 Estimates with moduli ?2? and ?2?.- 2.3 Estimates with modulus ?2d.- 2.4 Estimates with modulus ?2dd.- 2.5 Estimates with Ditzian-Totik modulus.- 3 Absolute Optimal Constants.- 3.1 Introduction.- 3.2 Discrete functionals and the classical second order modulus ?2.- 3.3 General functionals and the second order modulus with parameter ?2?.- 4 Estimates for the Bernstein Operators.- 4.1 Various types of estimates.- 4.2 Best constant in the estimate with modulus ?2.- 4.3 Global smoothness preservation.- 5 Two Classes of Bernstein Type Operators.- 5.1 Generalized Brass type operators.- 5.2 Generalized Durrmeyer type operators.- 6 Approximation Operators for Vector-Valued Functions.- 6.1 Approximation of functions with real argument.- 6.2 Approximation of functions with vector argument.- References.
Details
Erscheinungsjahr: 2004
Fachbereich: Analysis
Genre: Mathematik
Rubrik: Naturwissenschaften & Technik
Medium: Taschenbuch
Inhalt: x
202 S.
ISBN-13: 9780817643508
ISBN-10: 0817643508
Sprache: Englisch
Herstellernummer: 10996522
Ausstattung / Beilage: Paperback
Einband: Kartoniert / Broschiert
Autor: Paltanea, Radu
Herausgeber: George A Anastassiou
Hersteller: Birkh„user Boston
Birkhäuser Boston
Maße: 235 x 155 x 12 mm
Von/Mit: Radu Paltanea
Erscheinungsdatum: 17.09.2004
Gewicht: 0,335 kg
Artikel-ID: 102423511
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