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

摘要

We establish a relation between Gauss quadrature formulas on the interval [-1,1] that approximate integrals of the form I_σ(F) = ∫_(-1)~(+1)F(x)σ(x)dx and Szego quadrature formulas on the unit circle of the complex plane that approximate integrals of the form (I-tilde)_ω(f) = ∫_(-π)~πf(e~(iθ)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(cosθ)|sinθ|, 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 σ(x) and orthogonal Szego polynomials with respect to ω(θ). Inclusion of Gauss-Lobatto and Gauss-Radau formulas is natural.