Szegö quadrature

Szegö quadrature formulas are the analogues on the unit circle in the complex plane of the Gauss quadrature formulas on an interval (cf. also Gauss quadrature formula). They approximate the integral

where and is a positive measure on , by a quadrature formula of the form

One cannot take the zeros of the Szegö polynomials as nodes (as in Gaussian formulas), because these are all in the open unit disc (cf. also Szegö polynomial). Therefore, the para-orthogonal polynomials are introduced as , where and . These are orthogonal to and have simple zeros, which are on . The Szegö quadrature formula then takes as nodes the zeros , , of , and as weights the Christoffel numbers

The result is a quadrature formula with a maximal domain of validity in the set of Laurent polynomials, i.e., the formula is exact for all trigonometric polynomials in , a space of dimension , which is the maximal dimension possible with a quadrature formula of this form.

The Szegö quadrature formulas were introduced in [a2]. The underlying ideas have been generalized from polynomials to rational functions. See [a1].

References

[a1]	A. Bultheel, P. González-Vera, E. Hendriksen, O. Njåstad, "Quadrature and orthogonal rational functions" J. Comput. Appl. Math. , 127 (2001) pp. 67–91 (Invited paper)
[a2]	W.B. Jones, O. Njåstad, W.J. Thron, "Moment theory, orthogonal polynomials, quadrature and continued fractions associated with the unit circle" Bull. London Math. Soc. , 21 (1989) pp. 113–152