Dedicated to Professor William B. Jones on the occasion of his 70th birthday ABSTRACT. We consider some of the relations that exist between real Szegö polynomials and certain para-orthogonal polynomials defined on the unit circle, which are again related 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

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... lynomials on [−1, 1]. In the case of any symmetric quadrature rule on [−1, 1], its associated quadrature rule on the unit circle is also given.