Szegö quadrature

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

\begin{equation*} I _ { \mu } ( f ) = \int _ { T } f ( t ) d \mu ( t ), \end{equation*}

where $\mathbf{T} = \{ z \in \mathbf{C} : | z | = 1 \}$ and $\mu$ is a positive measure on $\bf T$, by a quadrature formula of the form

\begin{equation*} I _ { n } ( f ) = \sum _ { k = 1 } ^ { n } \lambda _ { n k } f ( \xi _ { n k } ). \end{equation*}

One cannot take the zeros of the Szegö polynomials $\phi_n$ as nodes (as in Gaussian formulas), because these are all in the open unit disc $\mathbf D$ (cf. also Szegö polynomial). Therefore, the para-orthogonal polynomials are introduced as $Q _ { n } ( z , \tau ) = \phi _ { n } ( z ) + \tau \phi _ { n } ^ { * } ( z )$, where $\tau \in \mathbf{T}$ and $\phi _ { n } ^ { * } ( z ) = z ^ {n } \overline { \phi _ { n } ( 1 / \overline{z} ) }$. These are orthogonal to $\{ z , \ldots , z ^ { n - 1 } \}$ and have $n$ simple zeros, which are on $\bf T$. The Szegö quadrature formula then takes as nodes the zeros $\xi _ { n , k }$, $k = 1 , \dots , n$, of $Q _ { n } ( z , \tau )$, and as weights the Christoffel numbers

\begin{equation*} \lambda _ { n k } = \frac { 1 } { \sum _ { j = 0 } ^ { n - 1 } | \phi _ { j } ( \xi _ { n k } ) | ^ { 2 } } > 0. \end{equation*}

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 $\operatorname { span } \{ z ^ { - n - 1 } , \dots , z ^ { - 1 } , 1 , z , \dots , z ^ { n - 1 } \}$, a space of dimension $2 n - 1$, 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