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 Iσ(F)=∫−1+1F(x)σ(x) dx and Szegő quadrature formulas on the unit circle of the complex plane that approximate integrals of the form Ĩω(f)=∫−ππf(eiθ)ω(θ) dθ. The weight σ(x) is positive on [−1,1] while the weight ω(θ) is positive on [−π,π]. It is shown that if ω(θ)=σ(cos θ)|sin θ|, then there is an intimate relation between the Gauss and Szegő quadrature formulas. Moreover, as a side result we also obtain an easy derivation for relations between orthogonal polynomials with respect to σ(x) and orthogonal Szegő polynomials with respect to ω(θ). Inclusion of Gauss–Lobatto and Gauss–Radau formulas is natural.