Szegö quadrature formulas for certain Jacobi-type weight functions

@article{Daruis2002SzegQF,
  title={Szeg{\"o} quadrature formulas for certain Jacobi-type weight functions},
  author={Leyla Daruis and Pablo Gonz{\'a}lez-Vera and Olav Nj{\aa}stad},
  journal={Math. Comput.},
  year={2002},
  volume={71},
  pages={683-701}
}
In this paper we are concerned with the estimation of integrals on the unit circle of the form ∫02π f(eiθ)ω(θ)dθ by means of the so-called Szego quadrature formulas, i.e., formulas of the type Σj=1n λjf(xj) with distinct nodes on the unit circle, exactly integrating Laurent polynomials in subspaces of dimension as high as possible. When considering certain weight functions ω(θ) related to the Jacobi functions for the interval [-1, 1], nodes {xj}j=1n and weights {λj}j=1n in Szego quadrature… 
View via Publisher
ams.org
19 Citations
Background Citations
2

Tables from this paper

19 Citations

Citation Type

Citation Type

Quadrature formulas on the unit circle with prescribed nodes and maximal domain of validity
Matrix methods for quadrature formulas on the unit circle. A survey
Positive interpolatory quadrature formulas and para-orthogonal polynomials
Positive trigonometric quadrature formulas and quadrature on the unit circle
TLDR
It is shown that the nodes polynomial can be generated by a simple recurrence relation and as a byproduct interlacing properties of zeros of para-orthogonal polynomials are obtained.
Ju n 20 06 A matrix approach to the computation of quadrature formulas on the unit circle 1
In this paper we consider a general sequence of orthogonal Laurent polynomials on the unit circle and we first study the equivalences between recurrences for such families and Szegő’s recursion and…
Anti-Szego quadrature rules
TLDR
This paper presents a new class of discretization methods, which it is referred to as anti-Szego quadrature rules, which can be used to estimate the error in Szego Quadrature Rules.
A connection between quadrature formulas on the unit circle and the interval [ - 1,1]
A matrix approach to the computation of quadrature formulas on the unit circle
Rational Szego quadratures associated with Chebyshev weight functions
TLDR
Rational Szegýo quadrature formulas associated with Chebyshev weight functions are characterized by giving explicit expressions for the corresponding para-orthogonal rational functions and weights in the quadratures.
Rational quadrature formulae on the unit circle with arbitrary poles
TLDR
Interpolatory quadrature rules exactly integrating rational functions on the unit circle are considered and a computable upper bound of the error is obtained which is valid for any choice of poles, arbitrary weight functions and any degree of exactness.
...
...

References

SHOWING 1-10 OF 12 REFERENCES
Bounds for remainder terms in Szego¨ quadrature on the unit circle
This paper deals with Szego quadrature for integration around the unit circle in the complex plane. Nodes for the quadrature formulas are the zeros ζ j (n) ( w n), j = 1,2,…, n, of para-orthogonal…
Szegö polynomials and quadrature formulas on the unit circle
A special class of polynomials orthogonal on the unit circle including the associated polynomials
Let (Pν) be a sequence of monic polynomials orthogonal on the unit circle with respect to a nonnegative weight function, let (Ωυ) the monic associated polynomials of (Pv), and letA andB be…
Characterization of orthogonal polynomials with respect to a functional
Algorithms to Compute the Reflection Coefficients of Digital Filters
TLDR
Three algorithms are given which compute explicitly or implicitly certain coefficients in a recursive scheme to generate polynomials and the Levinson and Schur methods are obtained to compute the reflection coefficients in linear predictive filtering.
Moment Theory, Orthogonal Polynomials, Quadrature, and Continued Fractions Associated with the unit Circle
Etude des relations entre la theorie des moments, les polynomes orthogonaux, la quadrature et les fractions continues associees au cercle-unite. Revue d'ensemble des resultats generaux
Some results about numerical quadrature on the unit circle
In this paper, quadrature formulas on the unit circle are considered. Algebraic properties are given and results concerning error and convergence established.Finally, numerical experiments are…
The Canary Islands
Now that the Canary Islands are rapidly becoming better known as one of the most advantageous health-resorts within easy reach of England, it may be of some interest to mention a few facts concerning…
Gautschi and G . V . Milovanović . Polynomials orthogonal on the semicircle
  • Journal of Approximation Theory
  • 1986
Semiclassical orthogonal polynomials on the unit circle. MAPA 3072
  • Special topics in Approximation Theory. Unpublished Technical Report
  • 1999
...
...