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Para-orthogonal rational matrix-valued functions on the unit circle. (English) Zbl 1272.47022

For a given sequence \(\{\alpha_0,\alpha_1,\alpha_2,\ldots\}\) (\(\alpha_0=0\)) of complex numbers, a nested sequence of spaces of rational functions \[ \mathcal{L}_n=\left\{{p_n(z)\over \pi_n(z)}: \pi_n(z)= \prod_{i=0}^n(1-\overline{\alpha}_iz), p_n(z)=\sum_{i=0}^n a_k z^k\right\},~~n=0,1,\ldots \] is considered. In [A. Bultheel et al., Orthogonal rational functions. Cambridge: Cambridge University Press (1999; Zbl 0923.42017)], the orthogonal polynomials on the unit circle have been generalized to orthogonal rational functions (ORFs) in this rational setting. Whereas the \(n\)th ORF is orthogonal to \(\mathcal{L}_{n-1}\), the \(n\)th para-ORF (PORF) is orthogonal to the subspace \(\{r\in\mathcal{L}_{n-1}: r(\alpha_n)=0\}\). The PORFs are important because they deliver simple zeros lying on the unit circle which can be used as nodes in a rational Szegő quadrature formula. There is also a connection between PORFs and the reproducing kernels for the spaces \(\mathcal{L}_n\).
In a sequence of papers by B. Fritzsche, B. Kirstein and coworkers, the theory of ORFs has been extended to the (square and rectangular) matrix valued case. The present paper is another one in this sequence related to the properties of (square matrix valued) PORFs. One has to deal with a left and a right version of a matrix valued inner product, and matrices that can be singular without being zero. Conditions for the existence of PORFs are given, generalizing a scalar version from [A. Bultheel and A. Lasarow, Analysis, München 30, No. 3, 301–316 (2010; Zbl 1257.42036)]. The paper arrives at a result about the zeros of the (determinants of) matrix valued ORFs. Here, the matrix case is clearly considerably more complicated than the scalar version.

MSC:

47A56 Functions whose values are linear operators (operator- and matrix-valued functions, etc., including analytic and meromorphic ones)
42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis
30E05 Moment problems and interpolation problems in the complex plane

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