Zeros of para–orthogonal polynomials and linear spectral transformations on the unit circle

@article{Castillo2015ZerosOP,
  title={Zeros of para–orthogonal polynomials and linear spectral transformations on the unit circle},
  author={Kenier Castillo and Francisco Marcell{\'a}n and Maria Das Neves Rebocho},
  journal={Numerical Algorithms},
  year={2015},
  volume={71},
  pages={699-714}
}
We study the interlacing properties of zeros of para–orthogonal polynomials associated with a nontrivial probability measure supported on the unit circle dµ and para–orthogonal polynomials associated with a modification of dµ by the addition of a pure mass point, also called Uvarov transformation. Moreover, as a direct consequence of our approach, we present some results related with the Christoffel transformation. 
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References

SHOWING 1-10 OF 64 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
Interlacing of zeros of shifted sequences of one-parameter orthogonal polynomials
We study the interlacing property of zeros of Laguerre polynomials of adjacent degree, where the free parameters differ by an integer, and of the same degree, where the free parameter is shifted
Generators of rational spectral transformations for nontrivial C-functions
TLDR
It is shown that a rational spectral transformation of FL with polynomial coefficients is a finite composition of four canonical spectral transformations.
Sobolev-type orthogonal polynomials on the unit ball
Orthogonal Polynomials and Rational Modifications of Measures
Abstract Given a finite positive measure on the Borel subsets of the complex plane with compact support containing infinitely many points, we deduce some formulas for the sequence of monic orthogonal
Rational spectral transformations and orthogonal polynomials
...
1
2
3
4
5
...