Algebraic properties of robust Pade ́ approximants

Dedicated to the memory of our friend Herbert Stahl and our colleague A.A. Gonchar. For a recent new numerical method for computing so-called robust Pade ́ approximants through SVD techniques, the authors gave numerical evidence that such approximants are insensitive to perturbations in the data, and do not have so-called spurious poles, that is, poles with a close-by zero or poles with small residuals. A black box procedure for eliminating spurious poles would have a major impact on the convergence theory of Pade ́ approximants since it is known that convergence in capacity plus absence of poles in some domain D implies locally uniform convergence in D. In the present paper we provide a proof for forward stability (or robustness), and show absence of spurious poles for the subclass of so-called well-conditioned Pade ́ approximants. We also give a numerical example of some robust Pade ́ approximant which has spurious poles, and discuss related questions. It turns out that it is not sufficient to discuss only linear algebra properties of the underlying rectangular Toeplitz matrix, since in our results other matrices like Sylvester matrices also occur. These types of matrices have been used before in numerical greatest common divisor computations. Key words: Pade ́ approximation, SVD, regularization, Froissart doublet, spurious poles