Positive trigonometric quadrature formulas and quadrature on the unit circle

@article{Peherstorfer2011PositiveTQ,
  title={Positive trigonometric quadrature formulas and quadrature on the unit circle},
  author={Franz Peherstorfer},
  journal={Math. Comput.},
  year={2011},
  volume={80},
  pages={1685-1701}
}
We give several descriptions of positive quadrature formulas which are exact for trigonometric -, respectively, Laurent polynomials of degree less or equal $n-1-m$, $0\leq m\leq n-1$. A complete and simple description is obtained with the help of orthogonal polynomials on the unit circle. In particular it is shown that the nodes polynomial can be generated by a simple recurrence relation. As a byproduct interlacing properties of zeros of para-orthogonal polynomials are obtained. Finally… 
TRIGONOMETRIC MULTIPLE ORTHOGONAL POLYNOMIALS OF SEMI-INTEGER DEGREE AND THE CORRESPONDING QUADRATURE FORMULAS
An optimal set of quadrature formulas with an odd number of nodes for trigonometric polynomials in Borges’ sense [Numer. Math. 67 (1994), 271-288], as well as trigonometric multiple orthogonal
Zeros of quasi-paraorthogonal polynomials and positive quadrature
Error estimates for some quadrature rules with maximal trigonometric degree of exactness
In this paper, we give error estimates for quadrature rules with maximal trigonometric degree of exactness with respect to an even weight function on ( − π,π) for integrand analytic in a certain
Zeros of para–orthogonal polynomials and linear spectral transformations on the unit circle
TLDR
The interlacing properties of zeros of para–orthogonal polynomials associated with a nontrivial probability measure supported on the unit circle dµ are studied and some results related with the Christoffel transformation are presented.
...
1
2
3
...

References

SHOWING 1-10 OF 39 REFERENCES
On bi-orthogonal systems of trigonometric functions and quadrature formulas for periodic integrands
In this paper, quadrature formulas with an arbitrary number of nodes and exactly integrating trigonometric polynomials up to degree as high as possible are constructed in order to approximate
Positive quadrature formulas III: asymptotics of weights
TLDR
For any polynomial t n which generates a positive qf, a weight function is given with respect to which t n is orthogonal to P n-1 and an asymptotic representation of the quadrature weights is derived.
Linear combinations of orthogonal polynomials generating positive quadrature formulas
Let pk(x) = x +■■■ , k e N0 , be the polynomials orthogonal on [-1, +1] with respect to the positive measure da . We give sufficient conditions on the real numbers p , j = 0, ... , m , such that the
Szegö quadrature formulas for certain Jacobi-type weight functions
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
Quadrature formula and zeros of para-orthogonal polynomials on the unit circle
Given a probability measure μ on the unit circle T, we study para-orthogonal polynomials Bn(.,w) (with fixed w ∈ T) and their zeros which are known to lie on the unit circle. We focus on the
The zeros of linear combinations of orthogonal polynomials
Characterization of Positive Quadrature Formulas
We give a complete description of those numerical integration formulas based on n nodes which have positive weights and are exact for polynomials of degree equal or less than $2n - 1 - m$, where $0
Characterization of Quadrature Formula II
In a recent paper (SIAM J. Math. Anal., 12 (1981), pp. 935–942) we have described positive quadrature formulas (qf). The purpose of this note is to complete our results on positive qf and to extend
...
1
2
3
4
...