Meromorphic approximants to complex Cauchy transforms with polar singularities

L. Baratchart ; M. L. Yattselev

Sbornik. Mathematics, Tome 200 (2009) no. 9, pp. 1261-1297

Résumé

We study AAK-type meromorphic approximants to functions of the form $$ F(z)=\int\frac{d\lambda(t)}{z-t}+R(z), $$ where $R$ is a rational function and $\lambda$ is a complex measure with compact regular support included in $(-1,1)$, whose argument has bounded variation on the support. The approximation is understood in the $L^p$-norm of the unit circle, $p\geqslant2$. We dwell on the fact that the denominators of such approximants satisfy certain non-Hermitian orthogonal relations with varying weights. They resemble the orthogonality relations that arise in the study of multipoint Padé approximants. However, the varying part of the weight implicitly depends on the orthogonal polynomials themselves, which constitutes the main novelty and the main difficulty of the undertaken analysis. We obtain that the counting measures of poles of the approximants converge to the Green equilibrium distribution on the support of $\lambda$ relative to the unit disc, that the approximants themselves converge in capacity to $F$, and that the poles of $R$ attract at least as many poles of the approximants as their multiplicity and not much more. Bibliography: 35 titles.

Détail

Keywords: meromorphic approximation, AAK-theory, rational approximation, non-Hermitian orthogonality, Hardy spaces, critical points.
Mots-clés : orthogonal polynomials