Orthogonal Rational Functions

Introduction This monograph forms an introduction to the theory of orthogonal rational functions. The simplest way to see what we mean by orthogonal rational functions is to consider them as generalizations of orthogonal polynomials. There is not much confusion about the meaning of an orthogonal polynomial sequence. One says that f n g 1 n=0 is an orthogonal polynomial sequence if n is a polynomial of degree n and it is orthogonal to all polynomials of lower degree. Thus given some nite positive measure (with possibly complex support), one considers the Hilbert space L 2 () of square integrable functions which contains the polynomial subspaces P n , n = 0; 1; : : :. Then f n g 1 n=0 is an orthogonal polynomial sequence if n 2 P n n P n?1 and n ? P n?1. In particular, when the support of the measure is (part of) the real line or of the complex unit circle one gets the most widely studied cases of such general orthogonal polynomials. Such orthogonal polyno-mials appear of course in many diierent aspects of theoretical analysis and applications. The topics which are central in our generalization to rational functions are moment problems, quadrature formulas and classical problems of complex approximation in the complex plane. Polynomials can be seen as rational functions whose poles are all xed at innnity. For the orthogonal rational functions, we shall x a sequence of poles f k g 1 k=1 which, in principle, can be taken anywhere in the extended complex plane. Some of these k can be repeated, possibly an innnite number of times, or they could be innnite etc. However the sequence is xed once and for all and the order in which the k occur (possible repetitions included) is also given. This will then deene the n-dimensional spaces of rational functions L n which consist of all the rational functions of degree n whose poles are among 1 ; : : :; n (including possible repetitions). We then consider f n g 1 n=0 to be a sequence of orthogonal rational functions if n 2 L n n L n?1 and n ? L n?1. There are two possible generalizations, depending on whether one generalizes the poly-nomials orthogonal on the real line or the polynomials orthogonal on the unit circle. The diierence lies in the location of the nite poles which are introduced in …