Schur Functions and Orthogonal Polynomials on the Unit Circle

We apply a theorem of Geronimus to derive some new formulas connecting Schur functions with orthogonal polynomials on the unit circle. The applications include the description of the associated measures and a short proof of Boyd’s result about Schur functions. We also give a simple proof for the above mentioned theorem of Geronimus. 1. Schur functions In what follows we adopt the following notations: D def = {z ∈ C : |z| < 1}, D stands for the closure of D, and Z def = N ∪ {0}. In addition, B denotes the set of Schur functions, namely, B def = {f : f is analytic and |f | < 1 in D} . Similarly, C stands for the set of Carathéodory functions, that is, C def = {F : F is analytic and RF > 0 in D , F (0) = 1} . If f is a Schur function, then F (z) = 1 + z f(z) 1− z f(z) (1) is a Carathéodory function, and, vice versa, if F is a Carathéodory function, then f(z) = 1 z F (z)− 1 F (z) + 1 (2) 1991 Mathematics Subject Classification. 42C05, 30B70.