Matrix Uvarov Transformation on the Unit Circle: Asymptotic Properties

Abstract

Let \(\sigma \) be an \(l \times l\) Hermitian matrix measure supported on the unit circle. In this contribution, we study some algebraic and analytic properties of matrix orthogonal polynomials associated with the Uvarov matrix transformation of \(\sigma \) defined by

$$\begin{aligned} \mathrm{d}\sigma _{u_m}(z)=\mathrm{d}\sigma (z)+\sum _{j=1}^m\mathbf{M} _j\delta (z-\zeta _j), \end{aligned}$$

where \(\mathbf{M} _j\) is an \(l \times l\) positive definite matrix, \(\zeta _j\in \mathbb {C}\) with \(\zeta _j\ne \zeta _i\) and \(\delta \) is the Dirac matrix measure.

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Acknowledgements

We thank the anonymous referees for their useful comments and suggestions. They greatly contributed to improve the contents and presentation of the manuscript. The work of the third author was supported by México’s Consejo Nacional de Ciencia y Tecnología (Conacyt) Grant 287523.

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Correspondence to Luis E. Garza.

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Dueñas, H., Fuentes, E. & Garza, L.E. Matrix Uvarov Transformation on the Unit Circle: Asymptotic Properties. Bull. Malays. Math. Sci. Soc. (2020). https://doi.org/10.1007/s40840-020-00947-2

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Keywords

  • Matrix orthogonal polynomials on the unit circle
  • Uvarov matrix transformation
  • Relative asymptotics
  • Zeros

Mathematics Subject Classification

  • 33C45
  • 33D45
  • 42C05