Nonlinear Approximation Theory - Dietrich Braess

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Résumé :
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 branchesof 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.

Sommaire:
I. Preliminaries.- ? 1. Some Notation, Definitions and Basic Facts.- ? 2. A Review of the Characterization of Nearest Points in Linear and Convex Sets.- ? 3. Linear and Convex Chebyshev Approximation.- ?4. L1-Approximation and Gaussian Quadrature Formulas.- II. Nonlinear Approximation: The Functional Analytic Approach.- ?1. Approximative Properties of Arbitrary Sets.- ?2. Solar Properties of Sets.- ? 3. Properties of Chebyshev Sets.- III. Methods of Local Analysis.- ?1. Critical Points.- ?2. Nonlinear Approximation in Hilbert Spaces.- ? 3. Varisolvency.- ?4. Nonlinear Chebyshev Approximation: The Differentiable Case.- ?5. The Gauss-Newton Method.- IV. Methods of Global Analysis.- ?1. Preliminaries. Basic Ideas.- ?2. The Uniqueness Theorem for Haar Manifolds.- ?3. An Example with One Nonlinear Parameter.- V. Rational Approximation.- ?1. Existence of Best Rational Approximations.- ?2. Chebyshev Approximation by Rational Functions.- ?3. Rational Interpolation.- ?4. Pad? Approximation andMoment Problems.- ?5. The Degree of Rational Approximation.- ?6. The Computation of Best Rational Approximations.- VI. Approximation by Exponential Sums.- ?1. Basic Facts.- ?2. Existence of Best Approximations.- ?3. Some Facts on Interpolation and Approximation.- VII. Chebyshev Approximation by ?-Polynomials.- ?1. Descartes Families.- ?2. Approximation by Proper ?-Polynomials.- ?3. Approximation by Extended ?-Polynomials: Elementary Theory.- ?4. The Haar Manifold Gn\Gn?1.- ?5. Local Best Approximations.- ?6. Maximal Components.- ?7. The Number of Local Best Approximations.- VIII. Approximation by Spline Functions with Free Nodes.- ?1. Spline Functions with Fixed Nodes.- ?2. Chebyshev Approximation by Spline Functions with Free Nodes.- ?3. Monosplines of Least L?-Norm.- ?4. Monosplines of Least L1-Norm.- ?5. Monosplines of Least Lp-Norm.- Appendix. The Conjectures of Bernstein and Erd?s.