Orthogonal polynomials in the complex plane and on the real line

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