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Approximation theory --- Interpolation --- Numerical analysis --- Applied Mathematics

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This book presents a unique survey of solutions in algebraic approximation. Several results related with direct and converse theorems in the theory of approximation by algebraic polynomials in a finite interval are discussed. Some of these results are not collected in any other book. In addition, facts concerning trigonometric approximation that are necessary for motivation and comparisons are included. The selection of papers that are referenced and discussed document trends in polynomial approximation from the 1950s to the present day.   Algebraic Approximation: A Guide to Past and Current Solutions will be invaluable to anyone seeking to understand the evolution of ideas in algebraic approximation. Its extensive bibliographic character will help finding the correct references for a specific result.

Approximation theory. --- Approximation theory --- Civil & Environmental Engineering --- Mathematics --- Physical Sciences & Mathematics --- Engineering & Applied Sciences --- Algebra --- Operations Research --- Algebra. --- Theory of approximation --- Mathematics. --- Approximations and Expansions. --- Functional analysis --- Functions --- Polynomials --- Chebyshev systems --- Mathematical analysis --- Math --- Science

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The approximation of a continuous function by either an algebraic polynomial, a trigonometric polynomial, or a spline, is an important issue in application areas like computer-aided geometric design and signal analysis. This book is an introduction to the mathematical analysis of such approximation, and, with the prerequisites of only calculus and linear algebra, the material is targeted at senior undergraduate level, with a treatment that is both rigorous and self-contained. The topics include polynomial interpolation; Bernstein polynomials and the Weierstrass theorem; best approximations in the general setting of normed linear spaces and inner product spaces; best uniform polynomial approximation; orthogonal polynomials; Newton-Cotes , Gauss and Clenshaw-Curtis quadrature; the Euler-Maclaurin formula ; approximation of periodic functions; the uniform convergence of Fourier series; spline approximation,with an extensive treatment of local spline interpolation,and its application in quadrature. Exercises are provided at the end of each chapter.

Asymptotic expansions. --- Transformations (Mathematics). --- Mathematics --- Physical Sciences & Mathematics --- Mathematical Theory --- Approximation theory. --- Numerical analysis. --- Theory of approximation --- Mathematics. --- Computer science --- Computer mathematics. --- Applied mathematics. --- Engineering mathematics. --- Mathematics, general. --- Computational Mathematics and Numerical Analysis. --- Approximations and Expansions. --- Mathematics of Computing. --- Appl.Mathematics/Computational Methods of Engineering. --- Functional analysis --- Functions --- Polynomials --- Chebyshev systems --- Mathematical analysis --- Computer science. --- Mathematical and Computational Engineering. --- Engineering --- Engineering analysis --- Informatics --- Science --- Computer mathematics --- Discrete mathematics --- Electronic data processing --- Math --- Approximation theory --- Data processing. --- Study and teaching. --- Computer science—Mathematics.

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This second, revised and substantially extended edition of Approximations and Endomorphism Algebras of Modules reflects both the depth and the width of recent developments in the area since the first edition appeared in 2006. The new division of the monograph into two volumes roughly corresponds to its two central topics, approximation theory (Volume 1) and realization theorems for modules (Volume 2). It is a widely accepted fact that the category of all modules over a general associative ring is too complex to admit classification. Unless the ring is of finite representation type we must limit attempts at classification to some restricted subcategories of modules. The wild character of the category of all modules, or of one of its subcategories C, is often indicated by the presence of a realization theorem, that is, by the fact that any reasonable algebra is isomorphic to the endomorphism algebra of a module from C. This results in the existence of pathological direct sum decompositions, and these are generally viewed as obstacles to classification. In order to overcome this problem, the approximation theory of modules has been developed. The idea here is to select suitable subcategories C whose modules can be classified, and then to approximate arbitrary modules by those from C. These approximations are neither unique nor functorial in general, but there is a rich supply available appropriate to the requirements of various particular applications. The authors bring the two theories together. The first volume, Approximations, sets the scene in Part I by introducing the main classes of modules relevant here: the S-complete, pure-injective, Mittag-Leffler, and slender modules. Parts II and III of the first volume develop the key methods of approximation theory. Some of the recent applications to the structure of modules are also presented here, notably for tilting, cotilting, Baer, and Mittag-Leffler modules. In the second volume, Predictions, further basic instruments are introduced: the prediction principles, and their applications to proving realization theorems. Moreover, tools are developed there for answering problems motivated in algebraic topology. The authors concentrate on the impossibility of classification for modules over general rings. The wild character of many categories C of modules is documented here by the realization theorems that represent critical R-algebras over commutative rings R as endomorphism algebras of modules from C. The monograph starts from basic facts and gradually develops the theory towards its present frontiers. It is suitable both for graduate students interested in algebra and for experts in module and representation theory.

Modules (Algebra) --- Moduli theory. --- Approximation theory. --- Finite number systems --- Modular systems (Algebra) --- Algebra --- Finite groups --- Rings (Algebra) --- Theory of approximation --- Functional analysis --- Functions --- Polynomials --- Chebyshev systems --- Theory of moduli --- Analytic spaces --- Functions of several complex variables --- Geometry, Algebraic --- Approximations of Modules. --- Cotorsion Pair. --- E-Ring. --- Endomorphism Algebra. --- Filtration. --- Infinite Dimensional Tilting Theory. --- Prediction Principle.

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This book grew out of a course taught in the Department of Mathematics, Indian Institute of Technology, Delhi, which was tailored to the needs of the applied community of mathematicians, engineers, physicists etc., who were interested in studying the problems of mathematical physics in general and their approximate solutions on computer in particular. Almost all topics which will be essential for the study of Sobolev spaces and their applications in the elliptic boundary value problems and their finite element approximations are presented. Also many additional topics of interests for specific applied disciplines and engineering, for example, elementary solutions, derivatives of discontinuous functions of several variables, delta-convergent sequences of functions, Fourier series of distributions, convolution system of equations etc. have been included along with many interesting examples.

Theory of distributions (Functional analysis) --- Sobolev spaces --- Spaces, Sobolev --- Function spaces --- Distribution (Functional analysis) --- Distributions, Theory of (Functional analysis) --- Functions, Generalized --- Generalized functions --- Functional analysis --- Distribution Theory. --- Elliptic Boundary Value Problem. --- Finite Element Approximation. --- Sobolev Space.