Orthogonal Polynomials in the Complex Plane and on the Real Line

We present a short introduction into general orthogonal polynomials in the complex plane, with special attention for the real line and the unit circle. Most of the material is classical and available in di®erent textbooks (see the references for relevant literature). This introduction brings together the analysis in the complex plane, the real line and the unit circle, which should be useful for those initiating their research in this ̄eld. The emphasis is on extremal properties, location of zeros, recurrence relation, and quadrature rather than on asymptotic results. 1 Orthogonal polynomials in the complex plane 1.1 Preliminaries. Let 1 be a positive Borel measure in the complex plane and consider the Hilbert space L (1) of measurable functions Á for which 2 Z 2 jÁ(z)j d1(z) <1: In L (1) we have the inner product 2 Z hÁ;Ãi = Á(z)Ã(z) d1(z); Á;Ã 2 L (1): 2 Suppose Á ; Á ; Á ; : : : is a system of linearly independent functions in L (1). 0 1 2 2 Quite often it is much more convenient to transform this system of linearly independent functions into another system of linearly independent functions ' ; ' ; ' ; : : : 0 1 2 such that ' is a linear combination of the n+ 1 functions Á ;Á ; : : : ; Á and n 0 1 n Z h' ;' i = ' (z)' (z) d1(z) = 0; m6= n: n m n m This new system of functions is then said to be a system of orthogonal functions with respect to the measure 1. If moreover Z 2 2 k' k = j' (z)j d1(z) = 1; n ̧ 0; n n then ' are orthonormal functions. n 1991 Mathematics Subject Classi ̄cation. Primary 42C05; Secondary 33C45. Senior Research Associate of the Belgian National Fund for Scienti ̄c Research. c °0000 American Mathematical Society 0000-0000/00 $1.00 + $.25 per page