Difference between revisions of "Szegö quadrature"
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| + | 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|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|measure]] on $\bf T$, by a [[Quadrature formula|quadrature formula]] of the form | |
| − | + | \begin{equation*} I _ { n } ( f ) = \sum _ { k = 1 } ^ { n } \lambda _ { n k } f ( \xi _ { n k } ). \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 | + | 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|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|Christoffel numbers]] |
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| + | \begin{equation*} \lambda _ { n k } = \frac { 1 } { \sum _ { j = 0 } ^ { n - 1 } | \phi _ { j } ( \xi _ { n k } ) | ^ { 2 } } > 0. \end{equation*} | ||
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| + | 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 [[#References|[a2]]]. The underlying ideas have been generalized from polynomials to rational functions. See [[#References|[a1]]]. | The Szegö quadrature formulas were introduced in [[#References|[a2]]]. The underlying ideas have been generalized from polynomials to rational functions. See [[#References|[a1]]]. | ||
====References==== | ====References==== | ||
| − | <table>< | + | <table><tr><td valign="top">[a1]</td> <td valign="top"> 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)</td></tr><tr><td valign="top">[a2]</td> <td valign="top"> 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</td></tr></table> |
Latest revision as of 16:59, 1 July 2020
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 |
How to Cite This Entry:
Szegö quadrature. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Szeg%C3%B6_quadrature&oldid=23067
Szegö quadrature. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Szeg%C3%B6_quadrature&oldid=23067
This article was adapted from an original article by A. Bultheel (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article