Orthogonal polynomials—Constructive theory and applications

Abstract

We consider polynomials orthogonal with respect to some measure on the real line. A basic problem in the constructive theory of such polynomials is the determination of their three-term recurrence relation, given the measure in question. Depending on what is know about this measure, there are different ways to proceed. If, as is typical in applications in the physical sciences, one knows the measure only through moment information, the appropriate procedure is an algorithm that goes back to Chebyshev. The algorithm in effect implements the nonlinear map from the given moments (or modified moments) to the desired recursion coefficients. The numerical effectiveness of this procedure is determined in an essential way by the numerical condition of this map. It is known that the map is ill-conditioned in the case of ordinary moments. For modified moments, it may or may not be well-conditioned, the matter depending on certain properties of the given measure and the additional measure defining modified moments. A theorem to this effect will be given and illustrated on some typical examples. If more is known about the given measure, for example, if it is absolutely continuous and can be evaluated pointwise, then a procedure can be employed which is based on an observation of Stieltjes. Stieltjes remarked that the desired recursion coefficients can be successively built up by alternating between known inner product formulae for these coefficients and the initial sections of the recurrence relation already obtained. An effective implementation of this idea requires a suitable discretization of the inner product. This requirement, while possibly a weakness of the method, also accounts for its beauty, since it leaves room for imagination and ingenuity. Used skillfully, the discretized Stieltjes procedure is among the most widely applicable and effective methods for generating orthogonal polynomials. Some examples will be given to illustrate its use. Finally, we show how our newly acquired capability of generating nonstandard orthogonal polynomials can be used to solve some special problems in approximation theory and in the summation of slowly convergent series. A novel set of polynomials orthogonal on the semicircle will also be mentioned briefly in connection with Cauchy principal value integrals.