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Rational approximation techniques and frequency design: a Zolotarev problem and the Schur algorithm

Abstract : This thesis presents some rational approximation and optimization techniques with applications to the synthesis and identification of passive systems. In the first part, we study a Zolotarev-type problem: to maximize on some set of intervals the infimum of the modulus of a rational function of given degree, under the constraint that the modulus of this function is bounded by 1 on another set of intervals. We are first concerned with the existence and the characterization of the solutions to such a problem. Next, a Remes-type algorithm and a differential-correction-type algorithm are studied. The link with the synthesis of microwave filters is carried out in detail. In fact, the theory we present allows one to compute multiband filtering functions with respect to given specifications. From the practical viewpoint, some microwave filters have been designed using this theory, and their theoretical response is compared to the real one. In the second part, the Schur rational approximation of a Schur function is studied. A Schur function is an analytic function whose modulus is bounded by 1 in the unit disk. First, the multipoint Schur algorithm is presented. It gives a parametrization of all strictly Schur functions. Next, the link with orthogonal rational functions is developed via a Geronimus-type theorem. The latter allows us to prove some approximation properties, where the interpolation points may tend to the unit circle. In particular, a convergence in the Poincare metric is obtained thanks to an extension of a Szego-type theorem. A numerical study for the computation of the Schur approximants of given degree is also presented.
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Vincent Lunot. Rational approximation techniques and frequency design: a Zolotarev problem and the Schur algorithm. Numerical Analysis [math.NA]. Université de Provence - Aix-Marseille I, 2008. English. ⟨NNT : 2008AIX11011⟩. ⟨tel-00711860⟩

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