 
Summary: An Analogue for Szego Polynomials of the Clenshaw Algorithm
Gregory S. Ammar \Lambda William B. Gragg y Lothar Reichel zx
November 21, 1991
Abstract
Linear combinations of polynomials that are orthogonal with respect to an inner product
defined on (part of) the real axis are commonly evaluated by the Clenshaw algorithm. We
present an analogous algorithm for the evaluation of a linear combination
P n
j=0 ff j OE j of
polynomials OE j that are orthogonal with respect to an inner product defined on (part of)
the unit circle. The OE j are known as Szego polynomials, and find applications, e.g., in
signal processing. We also discuss how to express
P n
j=0 ff j OE j as a linear combination of
monomials.
Key Words: Clenshaw algorithm, Szego polynomial
1 Introduction
Let (t) be a distribution function with infinitely many points of increase in the interval
[\Gamma; ], and define for polynomials p and q the inner product on the unit circle
(p; q) := 1
