ROCKY MOUNTAIN JOURNAL OF MATHEMATICS Volume 33, Number 2, Summer 2003 SZEG ¨ O POLYNOMIALS: QUADRATURE RULES ON THE UNIT CIRCLE AND ON [-1, 1] R. BRESSAN, S.F. MENEGASSO AND A. SRI RANGA Dedicated to Professor William B. Jones on the occasion of his 70th birthday ABSTRACT. We consider some of the relations that exist between real Szeg¨o polynomials and certain para-orthogonal polynomials defined on the unit circle, which are again re- lated to certain orthogonal polynomials on [-1, 1] through the transformation x =(z 1/2 +z -1/2 )/2. Using these relations we study the interpolatory quadrature rule based on the zeros of polynomials which are linear combinations of the orthogonal polynomials on [-1, 1]. In the case of any symmetric quadra- ture rule on [-1, 1], its associated quadrature rule on the unit circle is also given. 1. Introduction. Let dν (z) be a Borel measure on the unit circle, i.e., ν (e iθ ) is real, bounded, non-decreasing with infinitely many points of increase in 0 ≤ θ ≤ 2π, and let μ m = C z m dν (z) be the associated moments. Then μ -m = μ m and T n = μ 0 μ 1 ··· μ n-1 μ -1 μ 0 ··· μ n-2 . . . . . . . . . . . . μ -n+1 μ -n+2 ··· μ 0 > 0 =(-1) ⌊n/2⌋ μ -n+1 μ -n+2 ··· μ 0 μ -n+2 μ -n+3 ··· μ 1 . . . . . . . . . . . . μ 0 μ 1 ··· μ n-1 =(-1) ⌊n/2⌋ H (-n+1) n , for n ≥ 1. Here ⌊n/2⌋ represents the integer part of n/2. In relation to the above moments T n are known as the Toeplitz determinants and H (-n+1) n are known as the Hankel determinants. This research was supported by grants from CNP q and FAPESP of Brazil. Received by the editors on September 1, 2001, and in revised form on March 26, 2002. Copyright c 2003 Rocky Mountain Mathematics Consortium 567