Zeros of quasi-paraorthogonal polynomials and positive quadrature

@article{Bultheel2022ZerosOQ,
  title={Zeros of quasi-paraorthogonal polynomials and positive quadrature},
  author={Adhemar Bultheel and Ruym{\'a}n Cruz-Barroso and Carlos D{\'i}az-Mendoza},
  journal={ArXiv},
  year={2022},
  volume={abs/2110.10460}
}
In this paper we illustrate that paraorthogonality on the unit circle T is the counterpart to orthogonality on R when we are interested in the spectral properties. We characterize quasiparaorthogonal polynomials on the unit circle as the analogues of the quasi-orthogonal polynomials on R. We analyze the possibilities of preselecting some of its zeros, in order to build positive quadrature formulas with prefixed nodes and maximal domain of validity. These quadrature formulas on the unit circle… 
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SHOWING 1-10 OF 58 REFERENCES
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…
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.
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.
Rank one perturbations and the zeros of paraorthogonal polynomials on the unit circle
We prove several results about zeros of paraorthogonal polynomials using the theory of rank one perturbations of unitary operators. In particular, we obtain new details on the interlacing of zeros…
A matrix approach to the computation of quadrature formulas on the unit circle
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 Szego's recursion and…
A connection between Szegő-Lobatto and quasi Gauss-type quadrature formulas
TLDR
New results on positive quadrature formulas with prescribed nodes for the approximation of integrals with respect to a positive measure supported on the unit circle are obtained and a characterization of their existence is presented.
A contribution to quasi-orthogonal polynomials and associated polynomials
In this paper, we study the quasi-orthogonality of polynomials and their associated polynomials. First we give a connection between these quasi-orthogonal polynomials and linear algebra. Then we use…
On quasi-orthogonal polynomials
Abstract Chihara [On quasi orthogonal polynomials, Proc. Amer. Math. Soc. 8 (1957) , 765–767] has shown that quasi-orthogonal polynomials satisfy a three-term recurrence relation with polynomial…
Szegő-type quadrature formulas
Abstract We consider the approximation of integrals with respect to measures supported on the unit circle by means of positive quadrature formulas with maximal domain of exactness and up to three…
Quadrature formulas on the unit circle with prescribed nodes and maximal domain of validity
TLDR
These are (n+m)-point formulas for which m nodes are fixed in advance, with m=1 and m=2 respectively, and which have a maximal domain of validity in the space of Laurent polynomials.
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