The Theory of Best Approximation and Functional Analysis

Results and problems in the modern theory of best approximation, in which the methods of functional analysis are applied in a consequent manner.

Author: Ivan Singer

Publisher: SIAM

ISBN: 1611970547

Category: Mathematics

Page: 102

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Results and problems in the modern theory of best approximation, in which the methods of functional analysis are applied in a consequent manner. This modern theory constitutes both a unified foundation for the classical theory of best approximation and a powerful tool for obtaining new results.

The Theory of Best Approximation and Functional Analysis

[160] B. SENDov, Certain questions in the theory of approximations of functions and sets in the Hausdorff metric, Uspehi Mat. Nauk, 24, 5(149) (1969), pp. 141–178. (In Russian.) , Approximation relative to Hausdorff distance, ...

Author: Ivan Singer

Publisher: SIAM

ISBN: 9780898710106

Category: Mathematics

Page: 95

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Presents results and problems in the modern theory of best approximation, in which the methods of functional analysis are applied in a consequent manner.

Approximation Theory and Functional Analysis

The complex rational functions, в}(c), are defined similarly with "m and "n replaced with P}(c) and Рp (Ф), the polynomials with complex coefficients. А сопр1ex function defined on [ 0, 1) still has exactly one best approximation from ...

Author:

Publisher: Elsevier

ISBN: 0080871461

Category: Mathematics

Page: 448

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Approximation Theory and Functional Analysis

Functional Analysis in Asymmetric Normed Spaces

... the Academy of the Socialist Republic of Romania, Bucharest; Springer-Verlag, New York-Berlin 1970. , The theory of best approximation and functional analysis, C. B. M. S. Regional Conference Series in Applied Mathematics, No. 13.

Author: Stefan Cobzas

Publisher: Springer Science & Business Media

ISBN: 9783034804783

Category: Mathematics

Page: 219

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An asymmetric norm is a positive definite sublinear functional p on a real vector space X. The topology generated by the asymmetric norm p is translation invariant so that the addition is continuous, but the asymmetry of the norm implies that the multiplication by scalars is continuous only when restricted to non-negative entries in the first argument. The asymmetric dual of X, meaning the set of all real-valued upper semi-continuous linear functionals on X, is merely a convex cone in the vector space of all linear functionals on X. In spite of these differences, many results from classical functional analysis have their counterparts in the asymmetric case, by taking care of the interplay between the asymmetric norm p and its conjugate. Among the positive results one can mention: Hahn–Banach type theorems and separation results for convex sets, Krein–Milman type theorems, analogs of the fundamental principles – open mapping, closed graph and uniform boundedness theorems – an analog of the Schauder’s theorem on the compactness of the conjugate mapping. Applications are given to best approximation problems and, as relevant examples, one considers normed lattices equipped with asymmetric norms and spaces of semi-Lipschitz functions on quasi-metric spaces. Since the basic topological tools come from quasi-metric spaces and quasi-uniform spaces, the first chapter of the book contains a detailed presentation of some basic results from the theory of these spaces. The focus is on results which are most used in functional analysis – completeness, compactness and Baire category – which drastically differ from those in metric or uniform spaces. The book is fairly self-contained, the prerequisites being the acquaintance with the basic results in topology and functional analysis, so it may be used for an introduction to the subject. Since new results, in the focus of current research, are also included, researchers in the area can use it as a reference text.

Duality for Nonconvex Approximation and Optimization

Detailed proofs of results are given, along with varied illustrations. While many of the results have been published in mathematical journals, this is the first time these results appear in book form.

Author: Ivan Singer

Publisher: Springer Science & Business Media

ISBN: 9780387283951

Category: Mathematics

Page: 356

View: 599

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The theory of convex optimization has been constantly developing over the past 30 years. Most recently, many researchers have been studying more complicated classes of problems that still can be studied by means of convex analysis, so-called "anticonvex" and "convex-anticonvex" optimizaton problems. This manuscript contains an exhaustive presentation of the duality for these classes of problems and some of its generalization in the framework of abstract convexity. This manuscript will be of great interest for experts in this and related fields.

Best Approximation in Inner Product Spaces

[3] Chebyshev sets and the shapes of convex bodies, in Methods of Functional Analysis in Approximation Theory (edited by C. A. Micchelli, D. V. Pai, and B. V. Limaye), Birkhäuser Verlag, Boston, 1986, 97–121.

Author: Frank R. Deutsch

Publisher: Springer Science & Business Media

ISBN: 9781468492989

Category: Mathematics

Page: 338

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This is the first systematic study of best approximation theory in inner product spaces and, in particular, in Hilbert space. Geometric considerations play a prominent role in developing and understanding the theory. The only prerequisites for reading the book is some knowledge of advanced calculus and linear algebra.

Constructive Aspects of Functional Analysis

I. Singer Introduction Here we want to present briefly some results, problems and directions of research in the modern theory of best approximation, i.e. in which the methods of functional analysis are applied in a consequent manner.

Author: Giuseppe Geymonat

Publisher: Springer Science & Business Media

ISBN: 9783642109843

Category: Mathematics

Page: 854

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A. Balakrishnan: A constructive approach to optimal control.- R. Glowinski: Méthodes itératives duales pour la minimisation de fonctionnelles convexes.- J.L. Lions: Approximation numérique des inéquations d’évolution.- G. Marchuk: Introduction to the methods of numerical analysis.- U. Mosco: An introduction to the approximate solution of variational inequalities.- I. Singer: Best approximation in normed linear spaces.- G. Strang: A Fourier analysis of the finite element variational method.- M. Zerner: Caractéristiques d’approximation des compacts dans les espaces fonctionnels et problèmes aux limites elliptiques.

Approximation Theory in Tensor Product Spaces

T.J. Rivlin and R.J. Sibner, The degree of approzimation of certain functions of two variables by the sum of functions of ... I. Singer, “The Theory of Best Approximation and Functional Analysis,” Society for Industrial and Appl. Math., ...

Author: William A. Light

Publisher: Springer

ISBN: 9783540397410

Category: Mathematics

Page: 158

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Nonlinear Analysis

This book highlights interesting aspects of nonlinear analysis and optimization together with many applications in the areas of physical and social sciences including engineering.

Author: Qamrul Hasan Ansari

Publisher: Springer

ISBN: 9788132218838

Category: Mathematics

Page: 352

View: 829

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Many of our daily-life problems can be written in the form of an optimization problem. Therefore, solution methods are needed to solve such problems. Due to the complexity of the problems, it is not always easy to find the exact solution. However, approximate solutions can be found. The theory of the best approximation is applicable in a variety of problems arising in nonlinear functional analysis and optimization. This book highlights interesting aspects of nonlinear analysis and optimization together with many applications in the areas of physical and social sciences including engineering. It is immensely helpful for young graduates and researchers who are pursuing research in this field, as it provides abundant research resources for researchers and post-doctoral fellows. This will be a valuable addition to the library of anyone who works in the field of applied mathematics, economics and engineering.