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AuthorPฤ{131}ltฤ{131}nea, Radu. author
TitleApproximation Theory Using Positive Linear Operators [electronic resource] / by Radu Pฤ{131}ltฤ{131}nea
ImprintBoston, MA : Birkhรคuser Boston, 2004
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Descript X, 202 p. online resource


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. In this book, the crucial role of the second order moduli of continuity in the study of such operators is emphasized. New and efficient methods, applicable to general operators and to diverse concrete moduli, are presented. The advantages of these methods consist in obtaining improved and even optimal estimates, as well as in broadening the applicability of the results. Additional Topics and Features: * Examination of the multivariate approximation case * Special focus on the Bernstein operators, including applications, and on two new classes of Bernstein-type operators * Many general estimates, leaving room for future applications (e.g. the B-spline case) * Extensions to approximation operators acting on spaces of vector functions * Historical perspective in the form of previous significant 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


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 Ditzianโ{128}{148}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

Mathematics Algebra Field theory (Physics) Approximation theory Functional analysis Integral transforms Operational calculus Operator theory Applied mathematics Engineering mathematics Mathematics Approximations and Expansions Field Theory and Polynomials Functional Analysis Integral Transforms Operational Calculus Operator Theory Applications of Mathematics


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