Optimal Approximation Methods

In this chapter we outline a theory of optimal computational methods for general, nonlinear approximation problems. We define the notions of optimal algorithms and information, and we analyze the classes of parallel and sequential methods. We put special emphasis on linear problems as well as linear and spline algorithms. Several relationships between optimal methods and n-widths and s-numbers are also exhibited.

Applications

We shall now take time to demonstrate some applications of the foregoing material. We discuss here Burgers’ equation problem, the simplest fluid dynamics conservation law problem, as well as the approximation of band-limited signals, and the bisection method for nonlinear zero finding problems. In this section we illustrate the application of Sine approximation to the approximate solution of Burgers’ equation.

Explicit Sinc-Like Methods

In this section we derive several methods of approximation using the function values {f(kh)}∞k=- ∞ . We present a family of simple rational functions, which make possible the explicit and arbitrarily accurate rational approximation of the filter, the step (Heaviside) and the impulse (delta) functions. The chief advantage of these methods is that they make it possible to write down a simple and explicit rational approximation corresponding to any desired accuracy. Also, the three families of approximations are very simply connected with one another—the filter being related to the Heaviside via an elementary transformation, and the impulse being the derivative of the Heaviside. Thus, these methods make it possible for us to approximate generalized functions. In this section we discuss various methods, some of which are new, for approximating a function f ( t ) using the values f(0), f(±h), f(±2h), . .., where h > 0.

Sinc Approximation

Sine methods are a new family of self-contained methods of approximation, which have several advantages over classical methods of approximation in the case of the presence of end-point singularities, in the case when we have a semi-infinite or infinite interval of approximation, or in the case of the presence of a boundary layer situation.

Splines

Spline functions are important approximation tools in numerous applications for which high degree polynomial methods perform poorly, such as in computer graphics and geometric modelling, as well as for various engineering problems—especially those involving graphing of numerical solutions and noisy data. Algorithms based on spline functions enjoy minimal approximation errors in wide classes of problems and minimal complexity bounds. In this Chapter we provide a brief introduction to basic classes of polynomial splines, B-Splines, and abstract splines. Further study of spline algorithms as applied to linear problems is outlined in Chapter 7. In this section we define polynomial spline functions, exhibit their interpolatory properties, and construct algorithms to compute them. It turns out that these splines provide interpolating curves that do not exhibit the large oscillations associated with high degree interpolatory polynomials. This is why they find applications in univariate curve matching in computer graphics.

Classical Approximation

In this chapter we acquaint the reader with the theory of approximation of elements of normed spaces by elements of their finite dimensional subspaces. The theory of best approximation was originated between 1850 and 1860 by Chebyshev. His results and ideas have been extended and complemented in the 20th century by other eminent mathematicians, such as Bernstein, Jackson, and Kolmogorov. Initially, we present the classical theory of best approximation in the setting of normed spaces. Next, we discuss best approximation in unitary (inner product) spaces, and we present several practically important examples. Finally, we give a reasonably complete presentation of best uniform approximation, along with examples, the Remez algorithm, and including converse theorems about best approximation. The goal of this section is to present some general results on approximation in normed spaces.

Introduction to n-Widths and s-Numbers

n-Widths and s-numbers provide conceptual generalizations of the classical concepts of best approximation. These concepts give both a new idea of best approximation, and they also play an important role for better understanding of best approximation and complexity.

Moment Problems

In this chapter we present the classical moment problems as they have been mathematically defined. Moment problems are the simplest way to describe inverse problems mathematically. These problems were originally posed with moments being integrals of monomials. Such moment problems are ill-posed, and present considerable computational difficulty. On the other hand, moment problems whose moments are integrals of orthogonal bases can computationally be dealt with much more easily.