# Multipoint Schur's algorithm, rational orthogonal functions, asymptotic properties and Schur rational approximation

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Abstract : In a paper by Khrushchev, the connections between the Schur algorithm, the Wall's continued fractions and the orthogonal polynomials are revisited and used to establish some nice convergence properties of the sequence of Schur functions associated with a Schur function. In this report, we generalize some of Krushchev's results to the case of a multipoint Schur algorithm, that is a Schur algorithm where all the interpolation points are not taken in 0 but anywhere in the open unit disk. To this end, orthogonal rational functions and a recent generalization of Geronimus theorem are used. Then, we consider the problem of approximating a Schur function by a rational function which is also Schur. This problem of approximation is very important for the synthesis and identification of passive systems. We prove that all strictly Schur rational function of degree $n$ can be written as the $2n$-th convergent of the Schur algorithm if the interpolation points are correctly chosen. This leads to a parametrization using the multipoint Schur algorithm. Some examples are computed by an $L^2$ norm optimization process and the results are validated by comparison with the unconstrained $L^2$ rational approximation.
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Cited literature [23 references]

https://hal.inria.fr/inria-00311744
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Submitted on : Wednesday, August 20, 2008 - 4:46:16 PM
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• HAL Id : inria-00311744, version 1

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Vincent Lunot, Laurent Baratchart, Stanislas Kupin, Martine Olivi. Multipoint Schur's algorithm, rational orthogonal functions, asymptotic properties and Schur rational approximation. [Research Report] RR-6620, INRIA. 2008. ⟨inria-00311744⟩

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