Approximation by Polynomials

Abstract

This chapter introduces some of the essentials of approximation theory, in particular approximating functions by “nice” ones such as polynomials. In general, the intention of approximation theory is to replace some complicated function with a new function, one that is easier to work with, at the price of some (hopefully small) difference between the two functions. The new function is called an approximation. There are two crucial issues in using an approximation: first, how much simpler is the approximation? and second, how close is the approximation to the original function? Deciding which approximation to use requires an analysis of the trade-off between these two issues.

Of course, the answers to these two questions depend on the exact meanings of simpler and close, which vary according to the context. In this chapter, we study approximations by polynomials. Close is measured by some norm. We concentrate on the uniform norm, so that a polynomial approximation should be close to the function everywhere on a given interval.

Approximations are closely tied to the notions of limit and convergence, since a sequence of functions approximating a function f to greater and greater accuracy converges to f in the norm used. Different approximation schemes correspond to different norms.