# Critical Points and Error Rank in Best $H_2$ Matrix Rational Approximation of Fixed McMillan Degree

Abstract : This paper deals with best rational approximation of prescribed McMillan degree to matrix-valued functions in the real Hardy space of the complement of the unit disk endowed with the Frobenius $L_2$-norm. We describe the topological structure of the set of approximants in terms of inner--unstable factorizations. This allows us to establish a two--sided tangential interpolation equation for the critical points of the criterion, and to prove that the rank of the error $F-H$ is at most $k-n$ when $F$ is rational of degree $k$ and $H$ is critical of degree $n$. In the particular case where $k=n$, it follows that $H=F$ is the unique critical point, and this entails a local uniqueness result when approximating near--rational functions.
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Type de document :
Rapport
RR-2970, INRIA. 1996
Domaine :

https://hal.inria.fr/inria-00073728
Contributeur : Rapport de Recherche Inria <>
Soumis le : mercredi 24 mai 2006 - 13:37:06
Dernière modification le : samedi 27 janvier 2018 - 01:31:32
Document(s) archivé(s) le : dimanche 4 avril 2010 - 23:56:19

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• HAL Id : inria-00073728, version 1

### Citation

Laurent Baratchart, Martine Olivi. Critical Points and Error Rank in Best $H_2$ Matrix Rational Approximation of Fixed McMillan Degree. RR-2970, INRIA. 1996. 〈inria-00073728〉

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