A connection between quadrature formulas on the unit circle and the interval [-1,1]

We establish a relation between Gauss quadrature formulas on the interval [- 1,1] that approximate integrals of the form I-sigma(F)= integral (+1)(-1) F(x)alpha (x)dx and Szego quadrature formulas on the unit circle of the complex plane that approximate integrals of the form (I) over tilde (omega)(f)= integral (pi)(-pi)f(e(i theta))omega(theta )d theta. The weight sigma (x) is positive on [ - 1,1] while the weight omega(theta) is positive on [ - pi, pi]. It is shown that if omega(theta) = sigma (cos theta)/sin theta/, then there is an intimate relation between the Gauss and Szego quadrature formulas. Moreover, as a side result we also obtain an easy derivation for relations between orthogonal polynomials with respect to a(x) and orthogonal Szego polynomials with respect to omega(theta). Inclusion of Gauss-Lobatto and Gauss-Radau formulas is natural. (C) 2001 Elsevier Science B.V. All rights reserved.