On a Christoffel Transformation for Matrix Measures Supported on the Unit Circle

Abstract

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

$$\begin{aligned} d\sigma _{c_m}(z)=W_m(z)^Hd\sigma (z)W_m(z), \end{aligned}$$

where \(W_m(z)=\prod _{j=1}^m(z\mathbf{I} -A_j)\) and \(A_j\) is a square matrix for \(j=1,\ldots ,m.\) Moreover, we study the relative asymptotics of the associated orthogonal matrix polynomials when \(\sigma _{c_m}\) satisfies a matrix condition in the diagonal case. Some illustrative examples are considered.

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Acknowledgements

We thank the anonymous referee for her/his valuable comments and suggestions. They have contributed greatly to improve our manuscript. The work of the third author has been supported by México’s Consejo Nacional de Ciencia y Tecnología (Conacyt) Grant 287523.

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Correspondence to L. E. Garza.

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Dueñas, H., Fuentes, E. & Garza, L.E. On a Christoffel Transformation for Matrix Measures Supported on the Unit Circle. Comput. Methods Funct. Theory (2020). https://doi.org/10.1007/s40315-020-00324-x

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Keywords

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

Mathematics Subject Classification

  • 42C05
  • 33C45