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Session III.7 - Special Functions and Orthogonal Polynomials

Wednesday, June 21, 17:30 ~ 18:00

A least squares analog to the Nuttall-Pommerenke theorem.

Laurent Baratchart

INRIA centre de l'Université de Nice, France   -   This email address is being protected from spambots. You need JavaScript enabled to view it.

In various contexts involving identification and design, the following least-squares substitute to multipoint Padé approximation became quite popular in recent years under the name of "vector fitting": given a holomorphic function $f$ and a set of points $z_1,\cdots,z_N$ in the complex plane, to find a rational function $p_m/q_n$ of type $(m,n)$ minimizing the criterion $\sum_{j=1}^N |q(z_j)f(z_j)-p(z_j)|^2$. This type of approximation involves non-classical orthogonality, and its behaviour is still fairly open.

We analyze here the classical Padé analog where one minimizes the $l^2$-norm of the first n+m+1 terms of the Taylor expansion of the linearized error at a point $z_0$. In particular, we prove that convergence in capacity prevails when $f$ is analytic on the complex plane minus a polar set; i.e., a set of logarithmic capacity zero, provided that $N\leq C(n+m)$. This least-square version of the Nutall-Pommerenke theorem also sheds light on the multipoint case.

Joint work with Paul Asensio.

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