Lectures on Complex Approximation

Lectures on Complex Approximation PDF Author: GAIER
Publisher: Springer Science & Business Media
ISBN: 1461248140
Category : Mathematics
Languages : en
Pages : 207

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Book Description
The theory of General Relativity, after its invention by Albert Einstein, remained for many years a monument of mathemati cal speculation, striking in its ambition and its formal beauty, but quite separated from the main stream of modern Physics, which had centered, after the early twenties, on quantum mechanics and its applications. In the last ten or fifteen years, however, the situation has changed radically. First, a great deal of significant exper~en tal data became available. Then important contributions were made to the incorporation of general relativity into the framework of quantum theory. Finally, in the last three years, exciting devel opments took place which have placed general relativity, and all the concepts behind it, at the center of our understanding of par ticle physics and quantum field theory. Firstly, this is due to the fact that general relativity is really the "original non-abe lian gauge theory," and that our description of quantum field in teractions makes extensive use of the concept of gauge invariance. Secondly, the ideas of supersymmetry have enabled theoreticians to combine gravity with other elementary particle interactions, and to construct what is perhaps the first approach to a more finite quantum theory of gravitation, which is known as super gravity.

Lectures on Complex Approximation

Lectures on Complex Approximation PDF Author: GAIER
Publisher: Springer Science & Business Media
ISBN: 1461248140
Category : Mathematics
Languages : en
Pages : 207

Get Book Here

Book Description
The theory of General Relativity, after its invention by Albert Einstein, remained for many years a monument of mathemati cal speculation, striking in its ambition and its formal beauty, but quite separated from the main stream of modern Physics, which had centered, after the early twenties, on quantum mechanics and its applications. In the last ten or fifteen years, however, the situation has changed radically. First, a great deal of significant exper~en tal data became available. Then important contributions were made to the incorporation of general relativity into the framework of quantum theory. Finally, in the last three years, exciting devel opments took place which have placed general relativity, and all the concepts behind it, at the center of our understanding of par ticle physics and quantum field theory. Firstly, this is due to the fact that general relativity is really the "original non-abe lian gauge theory," and that our description of quantum field in teractions makes extensive use of the concept of gauge invariance. Secondly, the ideas of supersymmetry have enabled theoreticians to combine gravity with other elementary particle interactions, and to construct what is perhaps the first approach to a more finite quantum theory of gravitation, which is known as super gravity.

Methods of Approximation Theory in Complex Analysis and Mathematical Physics

Methods of Approximation Theory in Complex Analysis and Mathematical Physics PDF Author: Andrei A. Gonchar
Publisher: Springer
ISBN: 3540477926
Category : Mathematics
Languages : en
Pages : 225

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Book Description
The book incorporates research papers and surveys written by participants ofan International Scientific Programme on Approximation Theory jointly supervised by Institute for Constructive Mathematics of University of South Florida at Tampa, USA and the Euler International Mathematical Instituteat St. Petersburg, Russia. The aim of the Programme was to present new developments in Constructive Approximation Theory. The topics of the papers are: asymptotic behaviour of orthogonal polynomials, rational approximation of classical functions, quadrature formulas, theory of n-widths, nonlinear approximation in Hardy algebras,numerical results on best polynomial approximations, wavelet analysis. FROM THE CONTENTS: E.A. Rakhmanov: Strong asymptotics for orthogonal polynomials associated with exponential weights on R.- A.L. Levin, E.B. Saff: Exact Convergence Rates for Best Lp Rational Approximation to the Signum Function and for Optimal Quadrature in Hp.- H. Stahl: Uniform Rational Approximation of x .- M. Rahman, S.K. Suslov: Classical Biorthogonal Rational Functions.- V.P. Havin, A. Presa Sague: Approximation properties of harmonic vector fields and differential forms.- O.G. Parfenov: Extremal problems for Blaschke products and N-widths.- A.J. Carpenter, R.S. Varga: Some Numerical Results on Best Uniform Polynomial Approximation of x on 0,1 .- J.S. Geronimo: Polynomials Orthogonal on the Unit Circle with Random Recurrence Coefficients.- S. Khrushchev: Parameters of orthogonal polynomials.- V.N. Temlyakov: The universality of the Fibonacci cubature formulas.

Topics in Complex Approximation

Topics in Complex Approximation PDF Author: George A. Anastassiou
Publisher:
ISBN: 9781908106216
Category : Approximation theory
Languages : en
Pages : 150

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Book Description
In this monograph we study quantitatively the order of simultaneous approximation and Voronovskaja type asymptotic results for complex Bernstein-Schurer, Kantorovich-Schurer and Bernstein-Durrmeyer polynomials related to analytic functions on compact disks. In this way the overconvergence phenomenon for Bernstein-Schurer and Bernstein-Durrmeyer polynomials is revealed. We continue with explicit quantitative estimates for the overconvergence in the complex plane of the partial sums of the Fourier-type expansions on [-1, 1] with respect to Chebyshev and Legendre orthogonal polynomials. Furthermore we obtain quantitative estimates in the overconvergence phenomenon for the classical and generalized singular integrals of Gauss-Weierstrass, Poisson-Cauchy and Picard on a strip. Furthermore we present Jackson type approximation results by generalizations of multi-complex Picard, Poisson-Cauchy and Gauss-Weierstrass singular integrals in terms of higher order moduli of smoothness on polydisks. It follows quantitative estimates in the overconvergence phenomenon on polystrips, for the weighted and non-weighted cases, for generalized multicomplex singular integrals of Picard, Poisson-Cauchy and Gauss-Weierstrass types. We establish basic results concerning the best approximation of vector-valued functions by generalized polynomials. The overconvergence of singular integrals is presented for the first time in book form. This monograph is intended for researchers, graduate students working in many areas of pure and applied mathematics -- P. 4 of cover.

Interpolation and Approximation

Interpolation and Approximation PDF Author: Philip J. Davis
Publisher: Courier Corporation
ISBN: 0486624951
Category : Mathematics
Languages : en
Pages : 418

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Book Description
Intermediate-level survey covers remainder theory, convergence theorems, and uniform and best approximation. Other topics include least square approximation, Hilbert space, orthogonal polynomials, theory of closure and completeness, and more. 1963 edition.

Approximation, Complex Analysis, and Potential Theory

Approximation, Complex Analysis, and Potential Theory PDF Author: Norair Arakelian
Publisher: Springer Science & Business Media
ISBN: 9401009791
Category : Mathematics
Languages : en
Pages : 275

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Book Description
Hermann Weyl considered value distribution theory to be the greatest mathematical achievement of the first half of the 20th century. The present lectures show that this beautiful theory is still growing. An important tool is complex approximation and some of the lectures are devoted to this topic. Harmonic approximation started to flourish astonishingly rapidly towards the end of the 20th century, and the latest development, including approximation manifolds, are presented here. Since de Branges confirmed the Bieberbach conjecture, the primary problem in geometric function theory is to find the precise value of the Bloch constant. After more than half a century without progress, a breakthrough was recently achieved and is presented. Other topics are also presented, including Jensen measures. A valuable introduction to currently active areas of complex analysis and potential theory. Can be read with profit by both students of analysis and research mathematicians.

Topics in Approximation Theory

Topics in Approximation Theory PDF Author: Harold S. Shapiro
Publisher: Springer
ISBN: 3540364978
Category : Mathematics
Languages : en
Pages : 283

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Book Description


Approximation Theory and Approximation Practice, Extended Edition

Approximation Theory and Approximation Practice, Extended Edition PDF Author: Lloyd N. Trefethen
Publisher: SIAM
ISBN: 1611975948
Category : Mathematics
Languages : en
Pages : 375

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Book Description
This is a textbook on classical polynomial and rational approximation theory for the twenty-first century. Aimed at advanced undergraduates and graduate students across all of applied mathematics, it uses MATLAB to teach the field’s most important ideas and results. Approximation Theory and Approximation Practice, Extended Edition differs fundamentally from other works on approximation theory in a number of ways: its emphasis is on topics close to numerical algorithms; concepts are illustrated with Chebfun; and each chapter is a PUBLISHable MATLAB M-file, available online. The book centers on theorems and methods for analytic functions, which appear so often in applications, rather than on functions at the edge of discontinuity with their seductive theoretical challenges. Original sources are cited rather than textbooks, and each item in the bibliography is accompanied by an editorial comment. In addition, each chapter has a collection of exercises, which span a wide range from mathematical theory to Chebfun-based numerical experimentation. This textbook is appropriate for advanced undergraduate or graduate students who have an understanding of numerical analysis and complex analysis. It is also appropriate for seasoned mathematicians who use MATLAB.

Shape-Preserving Approximation by Real and Complex Polynomials

Shape-Preserving Approximation by Real and Complex Polynomials PDF Author: Sorin G. Gal
Publisher: Springer Science & Business Media
ISBN: 0817647031
Category : Mathematics
Languages : en
Pages : 359

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Book Description
First comprehensive treatment in book form of shape-preserving approximation by real or complex polynomials in one or several variables Of interest to grad students and researchers in approximation theory, mathematical analysis, numerical analysis, Computer Aided Geometric Design, robotics, data fitting, chemistry, fluid mechanics, and engineering Contains many open problems to spur future research Rich and updated bibliography

Topics in Complex Approximation Theory

Topics in Complex Approximation Theory PDF Author: Brett Chenoweth
Publisher:
ISBN:
Category :
Languages : en
Pages : 120

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Book Description


Fundamentals of Approximation Theory

Fundamentals of Approximation Theory PDF Author: Hrushikesh Narhar Mhaskar
Publisher: CRC Press
ISBN: 9780849309397
Category : Mathematics
Languages : en
Pages : 580

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Book Description
The field of approximation theory has become so vast that it intersects with every other branch of analysis and plays an increasingly important role in applications in the applied sciences and engineering. Fundamentals of Approximation Theory presents a systematic, in-depth treatment of some basic topics in approximation theory designed to emphasize the rich connections of the subject with other areas of study. With an approach that moves smoothly from the very concrete to more and more abstract levels, this text provides an outstanding blend of classical and abstract topics. The first five chapters present the core of information that readers need to begin research in this domain. The final three chapters the authors devote to special topics-splined functions, orthogonal polynomials, and best approximation in normed linear spaces- that illustrate how the core material applies in other contexts and expose readers to the use of complex analytic methods in approximation theory. Each chapter contains problems of varying difficulty, including some drawn from contemporary research. Perfect for an introductory graduate-level class, Fundamentals of Approximation Theory also contains enough advanced material to serve more specialized courses at the doctoral level and to interest scientists and engineers.