TABLE OF CONTENTS In this chapter we shall consider how to construct and use orthogonal systems of polynomials. They occur frequently in the physical sciences, and are very useful in computation. Having introduced the general concepts and the instances most commonly found in numerical work, we apply the technique first to the least square approximation of functions by polynomials (comparing the results with the minimax approximation); and second, to the important integration technique known as gaussian quadrature.
Let w( x) be a non-negative function on the interval [ a, b], such that for any integer n ? 0 ? a b x n w( x) dx is integrable. We shall call it a weight function, because it may be used to give more emphasis to one part of the interval than another. Furthermore, we shall assume that w( x) is such that if g( x) is a non-negative continuous function on the interval [ a, b], then ? a b w( x) g( x) dx = 0 implies g( x) ? 0. Commonly used weight functions are shown below, with the corresponding intervals of integration, labelled by the conventional names of the polynomials they generate:
Define the inner product of two continuous functions with respect to w by
Then the corresponding euclidean norm
satisfies all the requirements for a norm (see 11.1). In particular,...
