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## Über dieses Buch

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.

## Inhaltsverzeichnis

### Chapter I. Preliminaries

Abstract
Unless otherwise specified, E will be a real or complex normed linear space. A complete normed linear space is called a Banach space. E’ is the dual space, i.e., the Banach space of continuous linear functionals, equipped with the norm $$\left\| l \right\| = \mathop {\sup }\limits_{\left\| x \right\| = 1} \left| {l\left( x \right)} \right|$$.
Dietrich Braess

### Chapter II. Nonlinear Approximation: The Functional Analytic Approach

Abstract
In this chapter, existence and uniqueness of nearest points are considered in a general setting in which only properties from functional analysis such as the different kinds of compactness, convexity, and characterizations of best approximation in terms of linear functionals are used. In this framework the influence of uniform convexity, smoothness or non-smoothness of the underlying normed linear space is considered.
Dietrich Braess

### Chapter III. Methods of Local Analysis

Abstract
In the local theory, we shall study the characterization and other properties of local best approximations. Local solutions will be determined to be global solutions only if this is possible by simple methods, i.e. without using topological methods. Specifically, we shall use results from the linear theory and from convex approximation, where the characterization of local solutions is performed via approximation on tangent sets. This involves replacing the nonlinear problem by a linearized one.
Dietrich Braess

### Chapter IV. Methods of Global Analysis

Abstract
In the cases where the best approximation is not unique, information about the number of nearest points is desired. Statements on that number (even on the number of local solutions) are global results, in contrast with the characterization of local best approximations, which by definition refers to the behaviour in a neighborhood of the element under consideration.
Dietrich Braess

### Chapter V. Rational Approximation

Abstract
The approximation of functions by rational expressions is important in different disciplines of analysis. First one thinks of the representation of special functions by rational approximations for the use in computers, but the applications go far beyond this point. First, rational approximations arise quite naturally in the numerical solution of ordinary and parabolic differential equations and in the study of other numerical methods. Furthermore, the Stieltjes and the Hamburger moment problem can be well understood via methods from rational approximation. The latter shows that nonlinear approximation theory may be helpful even for problems which originally are convex problems and not nonlinear in a strict sense.
Dietrich Braess

### Chapter VI. Approximation by Exponential Sums

Abstract
The rational functions and exponential sums belong to those concrete families of functions which are the most frequently studied in nonlinear approximation theory. The starting point in the consideration of exponential sums is an approximation problem often encountered for the analysis of decay processes in natural sciences. A given empirical function on a real interval is to be approximated by sums of the form
$$\sum\limits_{v = 1}^n {{\alpha _v}{e^{{t_v}x}}}$$
where the parameters α v and t v are to be determined, while n is fixed.
Dietrich Braess

### Chapter VII. Chebyshev Approximation by γ-Polynomials

Abstract
The approximation by sums of exponentials shares some of the properties of rational approximation. But there is an essential difference: the best approximation is not always unique. There may be more than one isolated solution, which means that phenomena arise which are not met in the linear theory. Fortunately, it is possible to establish explicit bounds for the number of solutions. In order to get them, the results for Haar embedded manifold (derived with methods of global analysis), are applied.
Dietrich Braess

### Chapter VIII. Approximation by Spline Functions with Free Nodes

Abstract
In the last two decades, approximation by spline functions has become an important tool in applied mathematics, since the accuracy of the approximation depends only on local properties of the given function. As a consequence, spline functions with fixed nodes lead only to weak Haar spaces and the interpolation is governed by an interlacing property between the nodes and the points of interpolation. Nevertheless, a best approximation can be characterized in terms of alternants (with a possible degenerate length).
Dietrich Braess

### Backmatter

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