Meromorphic Approximation: Symmetric Contours and Wandering Poles

This manuscript reviews the study of the asymptotic behavior of meromorphic approximants to classes of functions holomorphic at infinity. The asymptotic theory of meromorphic approximation is primarily concerned with establishing the types of convergence, describing the domains where this convergence takes place, and identifying its exact rates. As the first question is classical, it is the latter two topics that this survey is mostly focused on with the greater emphasis on the exact rates. Three groups of approximants are introduced: meromorphic (AAK-type) approximants, L2-best rational approximants, and rational interpolants with free poles. Despite the groups being distinctively different, they share one common feature: much of the information on their asymptotic behavior is encoded in the non-Hermitian orthogonality relations satisfied by the polynomials vanishing at the poles of the approximants with the weight of orthogonality coming from the approximated function. The main goal of the study is extracting the generic asymptotic behavior of the zeros of these polynomials from the orthogonality relations and tracking down those zeros that do not conform to the general pattern (wandering poles of the approximants).