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AuthorBraess, Dietrich. author
TitleNonlinear Approximation Theory [electronic resource] / by Dietrich Braess
ImprintBerlin, Heidelberg : Springer Berlin Heidelberg, 1986
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Descript XIV, 290p. 38 illus. online resource


The first investigations of nonlinear approximation problems were made by P.L. Chebyshev in the last century, and the entire theory of uniform approximaยญ 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 introยญ 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 probยญ 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 polyยญ nomials, is treated in the appendix of this book


I. Preliminaries -- ยง 1. Some Notation, Definitions and Basic Facts -- ยง 2. A Review of the Characterization of Nearest Points in Linear and Convex Sets -- ยง 3. Linear and Convex Chebyshev Approximation -- ยง4. L1-Approximation and Gaussian Quadrature Formulas -- II. Nonlinear Approximation: The Functional Analytic Approach -- ยง1. Approximative Properties of Arbitrary Sets -- ยง2. Solar Properties of Sets -- ยง 3. Properties of Chebyshev Sets -- III. Methods of Local Analysis -- ยง1. Critical Points -- ยง2. Nonlinear Approximation in Hilbert Spaces -- ยง 3. Varisolvency -- ยง4. Nonlinear Chebyshev Approximation: The Differentiable Case -- ยง5. The Gauss-Newton Method -- IV. Methods of Global Analysis -- ยง1. Preliminaries. Basic Ideas -- ยง2. The Uniqueness Theorem for Haar Manifolds -- ยง3. An Example with One Nonlinear Parameter -- V. Rational Approximation -- ยง1. Existence of Best Rational Approximations -- ยง2. Chebyshev Approximation by Rational Functions -- ยง3. Rational Interpolation -- ยง4. Padรฉ Approximation and Moment Problems -- ยง5. The Degree of Rational Approximation -- ยง6. The Computation of Best Rational Approximations -- VI. Approximation by Exponential Sums -- ยง1. Basic Facts -- ยง2. Existence of Best Approximations -- ยง3. Some Facts on Interpolation and Approximation -- VII. Chebyshev Approximation by ?-Polynomials -- ยง1. Descartes Families -- ยง2. Approximation by Proper ?-Polynomials -- ยง3. Approximation by Extended ?-Polynomials: Elementary Theory -- ยง4. The Haar Manifold Gn\Gn?1 -- ยง5. Local Best Approximations -- ยง6. Maximal Components -- ยง7. The Number of Local Best Approximations -- VIII. Approximation by Spline Functions with Free Nodes -- ยง1. Spline Functions with Fixed Nodes -- ยง2. Chebyshev Approximation by Spline Functions with Free Nodes -- ยง3. Monosplines of Least L?-Norm -- ยง4. Monosplines of Least L1-Norm -- ยง5. Monosplines of Least Lp-Norm -- Appendix. The Conjectures of Bernstein and Erdรถs

Mathematics Numerical analysis Mathematics Numerical Analysis


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