Approximation Theory

Approximation theory is an area of mathematics that has become indispensable to the computational sciences. The approximation of magnitudes and functions describing some physical behavior is an integral part of scientific computing, queueing problems, neural networks, graphics, robotics, network traffic, financial trading, antenna design, floating-point arithmetic, image processing, speech analysis, and video signal filtering, to name just a few areas. The idea of seeking a simple mathematical function that describes some behavior approximately has two sources of motivation. The exact behavior that one is studying may not be able to be expressed in a closed mathematical formula. But even if an exact description is available it may be far too complicated for practical use. In both cases a best and simple approximation is required. What is meant by best and simple depends on the application at hand. The approximation is often used in a computer implementation, and therefore its evaluation needs to be efficient. The simplest and fastest functions for implementation are polynomials, because they use only the fast hardware operations of addition and multiplication. Next come rational functions, which also need the hardware operation division, one or more depending on their representation as a quotient of polynomials or as a continued fraction. Rational functions offer the clear advantage that they can reproduce asymptotic behavior (vertical, horizontal, slant), which is something polynomials are incapable of doing. For periodic phenomena, linear combinations of trigonometric functions make good candidates. For growth models or decaying magnitudes, linear combinations of exponentials can be used. In approximation theory one distinguishes between interpolation and so-called least-squares problems. In the former one wants the approximate model to take exactly the same values as prescribed by data given at precise argument values. In the latter a set of data (not necessarily discrete) is regarded as a trend and approximated by a simple model in one or other best sense. The difference is formalized in the following sections. Besides constructing a good and efficient mathematical model, one should also take the following two issues into account.