Nonlinear Approximation Theory

Author : Dietrich Braess
Publisher : Springer Science & Business Media
Page : 305 pages
File Size : 49,7 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9783642616099

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Nonlinear Approximation Theory by Dietrich Braess Pdf

The first investigations of nonlinear approximation problems were made by P.L. Chebyshev in the last century, and the entire theory of uniform approxima tion is strongly connected with his name. By making use of his ideas, the theories of best uniform approximation by rational functions and by polynomials were developed over the years in an almost unified framework. The difference between linear and rational approximation and its implications first became apparent in the 1960's. At roughly the same time other approaches to nonlinear approximation were also developed. The use of new tools, such as nonlinear functional analysis and topological methods, showed that linearization is not sufficient for a complete treatment of nonlinear families. In particular, the application of global analysis and the consideration of flows on the family of approximating functions intro duced ideas which were previously unknown in approximation theory. These were and still are important in many branches of analysis. On the other hand, methods developed for nonlinear approximation prob lems can often be successfully applied to problems which belong to or arise from linear approximation. An important example is the solution of moment problems via rational approximation. Best quadrature formulae or the search for best linear spaces often leads to the consideration of spline functions with free nodes. The most famous problem of this kind, namely best interpolation by poly nomials, is treated in the appendix of this book.

Author : Qamrul Hasan Ansari
Publisher : Springer
Page : 362 pages
File Size : 54,6 Mb
Release : 2014-06-05
Category : Mathematics
ISBN : 9788132218838

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Nonlinear Analysis by Qamrul Hasan Ansari Pdf

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.

Author : Andrei A. Gonchar,Edward B. Saff
Publisher : Springer
Page : 225 pages
File Size : 40,6 Mb
Release : 2008-01-03
Category : Mathematics
ISBN : 9783540477921

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Methods of Approximation Theory in Complex Analysis and Mathematical Physics by Andrei A. Gonchar,Edward B. Saff Pdf

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.

Author : Zvi Ziegler
Publisher : Unknown
Page : 384 pages
File Size : 48,9 Mb
Release : 1981
Category : Mathematics
ISBN : UCAL:B4979657

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Approximation Theory and Applications by Zvi Ziegler Pdf

Construction of elements of the relative chebyshev center. The numerical claculation of spline approximations on a binfinite. Global analysis in nonlinear approximation and its application to exponential approximation. Global analysis in nonlinear approximation and its application to exponential approximation. Simultaneous approximation and restricted chebyshev centers in function spaces. Quelques proprietes D'Une family D'operateurs positfs sur des espaces de functions relles definies presque partout sur ... Bell-Shaped basis functions for surface fitting. The n-Widhts of sets of analytic functions. Admissibility of quadrature formulas with random nodes. Convergence for operators of hyperbolic type. Explicit ... - extensions of functions of two variebles in a strip between two curves, or in a corner in IR ... Taylor interpolation of order n at the vertices of a triangle. Applications for hermite interpolation and finite elements. Jacobi projections. Oscillating monosplines of least uniform norm. Some applications and drawbacks of padé approximants. From dirac distributions to multivariate representation formulas. A new iterative method for the solution of systems nonlinear equations. Polynomials and rational functions. Quadrature formulae based on shape preserving interpolation. Optimal recovery among the polynomials. On cardinal spline interpolants. Approximation by lacunary polynomials: A converse theorem. An interpolatory rational approximation. Design problems for optimal surface interpolation. Open problems.

Author : Lloyd N. Trefethen
Publisher : SIAM
Page : 375 pages
File Size : 49,9 Mb
Release : 2019-01-01
Category : Mathematics
ISBN : 9781611975949

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Approximation Theory and Approximation Practice, Extended Edition by Lloyd N. Trefethen Pdf

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.

Author : Mark French,Csaba Szepesvári,Eric Rogers
Publisher : John Wiley & Sons
Page : 428 pages
File Size : 43,6 Mb
Release : 2003-06-27
Category : Science
ISBN : 0471498092

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Performance of Nonlinear Approximate Adaptive Controllers by Mark French,Csaba Szepesvári,Eric Rogers Pdf

In recent years there has been a wide interest in non-linear adaptive control using approximate models, either for tracking or regulation, and usually under the banner of neural network based control. The authors present a unique critical evaluation of the approximate model philosophy and its setting, rigorously comparing the performance of such controls against competing designs. Analysing a very topical aspect of contemporary research and control practice this book highlights the situations in which approximate model based designs are most appropriate and indicates scenarios in which other designs could be used more productively. Throughout the text concepts are illustrated using a variety of examples, both academic problems and those based on physical examples. The work is designed to open the door to realistic applications. * Unified coverage of the theory and application of a wide range of control systems areas including neural network based control and control using the approximate model * Presents a mathematically well founded introduction to the area of intelligent control * A varied selecion of practical examples drawn from a variety of fields, including robotics and aerospace, illustrate theoretical principles * Clear compaisons of a variety of control designs * Cross disciplinary approach to this leading edge topic A valuable reference for control practitioners and theorists, artificial intelligence researchers and applied mathematicians, as well as graduate students and researchers with an interest in adaptive control and stability.

Author : A.A. Gonchar,E.B. Saff
Publisher : Springer Science & Business Media
Page : 463 pages
File Size : 47,5 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9781461229667

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Progress in Approximation Theory by A.A. Gonchar,E.B. Saff Pdf

Designed to give a contemporary international survey of research activities in approximation theory and special functions, this book brings together the work of approximation theorists from North America, Western Europe, Asia, Russia, the Ukraine, and several other former Soviet countries. Contents include: results dealing with q-hypergeometric functions, differencehypergeometric functions and basic hypergeometric series with Schur function argument; the theory of orthogonal polynomials and expansions, including generalizations of Szegö type asymptotics and connections with Jacobi matrices; the convergence theory for Padé and Hermite-Padé approximants, with emphasis on techniques from potential theory; material on wavelets and fractals and their relationship to invariant measures and nonlinear approximation; generalizations of de Brange's in equality for univalent functions in a quasi-orthogonal Hilbert space setting; applications of results concerning approximation by entire functions and the problem of analytic continuation; and other topics.

Author : S.P. Singh,B. Watson,P. Srivastava
Publisher : Springer Science & Business Media
Page : 231 pages
File Size : 48,9 Mb
Release : 2013-04-17
Category : Mathematics
ISBN : 9789401588225

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Fixed Point Theory and Best Approximation: The KKM-map Principle by S.P. Singh,B. Watson,P. Srivastava Pdf

The aim of this volume is to make available to a large audience recent material in nonlinear functional analysis that has not been covered in book format before. Here, several topics of current and growing interest are systematically presented, such as fixed point theory, best approximation, the KKM-map principle, and results related to optimization theory, variational inequalities and complementarity problems. Illustrations of suitable applications are given, the links between results in various fields of research are highlighted, and an up-to-date bibliography is included to assist readers in further studies. Audience: This book will be of interest to graduate students, researchers and applied mathematicians working in nonlinear functional analysis, operator theory, approximations and expansions, convex sets and related geometric topics and game theory.

Author : Ole Christensen,Khadija Laghrida Christensen
Publisher : Springer Science & Business Media
Page : 166 pages
File Size : 47,8 Mb
Release : 2012-11-04
Category : Mathematics
ISBN : 9780817644482

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Approximation Theory by Ole Christensen,Khadija Laghrida Christensen Pdf

This concisely written book gives an elementary introduction to a classical area of mathematics – approximation theory – in a way that naturally leads to the modern field of wavelets. The exposition, driven by ideas rather than technical details and proofs, demonstrates the dynamic nature of mathematics and the influence of classical disciplines on many areas of modern mathematics and applications. Featuring classical, illustrative examples and constructions, exercises, and a discussion of the role of wavelets to areas such as digital signal processing and data compression, the book is one of the few to describe wavelets in words rather than mathematical symbols.

Author : Maria Tomas-Rodriguez,Stephen P. Banks
Publisher : Springer Science & Business Media
Page : 303 pages
File Size : 54,7 Mb
Release : 2010-02-04
Category : Mathematics
ISBN : 9781849961004

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Linear, Time-varying Approximations to Nonlinear Dynamical Systems by Maria Tomas-Rodriguez,Stephen P. Banks Pdf

Linear, Time-varying Approximations to Nonlinear Dynamical Systems introduces a new technique for analysing and controlling nonlinear systems. This method is general and requires only very mild conditions on the system nonlinearities, setting it apart from other techniques such as those – well-known – based on differential geometry. The authors cover many aspects of nonlinear systems including stability theory, control design and extensions to distributed parameter systems. Many of the classical and modern control design methods which can be applied to linear, time-varying systems can be extended to nonlinear systems by this technique. The implementation of the control is therefore simple and can be done with well-established classical methods. Many aspects of nonlinear systems, such as spectral theory which is important for the generalisation of frequency domain methods, can be approached by this method.

Author : Simeon Reich,Alexander J. Zaslavski
Publisher : Springer Science & Business Media
Page : 529 pages
File Size : 44,5 Mb
Release : 2013-11-21
Category : Mathematics
ISBN : 9781461495338

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Genericity in Nonlinear Analysis by Simeon Reich,Alexander J. Zaslavski Pdf

This book presents an extensive collection of state-of-the-art results and references in nonlinear functional analysis demonstrating how the generic approach proves to be very useful in solving many interesting and important problems. Nonlinear analysis plays an ever-increasing role in theoretical and applied mathematics, as well as in many other areas of science such as engineering, statistics, computer science, economics, finance, and medicine. The text may be used as supplementary material for graduate courses in nonlinear functional analysis, optimization theory and approximation theory, and is a treasure trove for instructors, researchers, and practitioners in mathematics and in the mathematical sciences. Each chapter is self-contained; proofs are solid and carefully communicated. Genericity in Nonlinear Analysis is the first book to systematically present the generic approach to nonlinear analysis. Topics presented include convergence analysis of powers and infinite products via the Baire Category Theorem, fixed point theory of both single- and set-valued mappings, best approximation problems, discrete and continuous descent methods for minimization in a general Banach space, and the structure of minimal energy configurations with rational numbers in the Aubry–Mather theory.

Author : Ronald DeVore,Angela Kunoth
Publisher : Springer Science & Business Media
Page : 671 pages
File Size : 48,8 Mb
Release : 2009-09-16
Category : Mathematics
ISBN : 9783642034138

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Multiscale, Nonlinear and Adaptive Approximation by Ronald DeVore,Angela Kunoth Pdf

The book of invited articles offers a collection of high-quality papers in selected and highly topical areas of Applied and Numerical Mathematics and Approximation Theory which have some connection to Wolfgang Dahmen's scientific work. On the occasion of his 60th birthday, leading experts have contributed survey and research papers in the areas of Nonlinear Approximation Theory, Numerical Analysis of Partial Differential and Integral Equations, Computer-Aided Geometric Design, and Learning Theory. The main focus and common theme of all the articles in this volume is the mathematics building the foundation for most efficient numerical algorithms for simulating complex phenomena.

Author : George A. Anastassiou
Publisher : Unknown
Page : 128 pages
File Size : 48,7 Mb
Release : 2018
Category : Approximation theory
ISBN : 3319895109

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Nonlinearity by George A. Anastassiou Pdf

This book focuses on approximations under the presence of ordinary and fractional smoothness, presenting both the univariate and multivariate cases. It also explores approximations under convexity and a new trend in approximation theory -approximation by sublinear operators with applications to max-product operators, which are nonlinear and rational providing very fast and flexible approximations. The results presented have applications in numerous areas of pure and applied mathematics, especially in approximation theory and numerical analysis in both ordinary and fractional senses. As such this book is suitable for researchers, graduate students, and seminars of the above disciplines, and is a must for all science and engineering libraries.

Author : G. A. Watson
Publisher : John Wiley & Sons
Page : 248 pages
File Size : 49,6 Mb
Release : 1980
Category : Mathematics
ISBN : UOM:39015015733853

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Approximation Theory and Numerical Methods by G. A. Watson Pdf

Author : Hubertus Th. Jongen,P. Jonker,F. Twilt
Publisher : Springer Science & Business Media
Page : 516 pages
File Size : 50,5 Mb
Release : 2013-12-11
Category : Mathematics
ISBN : 9781461500179

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Nonlinear Optimization in Finite Dimensions by Hubertus Th. Jongen,P. Jonker,F. Twilt Pdf

At the heart of the topology of global optimization lies Morse Theory: The study of the behaviour of lower level sets of functions as the level varies. Roughly speaking, the topology of lower level sets only may change when passing a level which corresponds to a stationary point (or Karush-Kuhn Tucker point). We study elements of Morse Theory, both in the unconstrained and constrained case. Special attention is paid to the degree of differentiabil ity of the functions under consideration. The reader will become motivated to discuss the possible shapes and forms of functions that may possibly arise within a given problem framework. In a separate chapter we show how certain ideas may be carried over to nonsmooth items, such as problems of Chebyshev approximation type. We made this choice in order to show that a good under standing of regular smooth problems may lead to a straightforward treatment of "just" continuous problems by means of suitable perturbation techniques, taking a priori nonsmoothness into account. Moreover, we make a focal point analysis in order to emphasize the difference between inner product norms and, for example, the maximum norm. Then, specific tools from algebraic topol ogy, in particular homology theory, are treated in some detail. However, this development is carried out only as far as it is needed to understand the relation between critical points of a function on a manifold with structured boundary. Then, we pay attention to three important subjects in nonlinear optimization.