We constructively solve a pair of band-limited generalizations of the Adamjan-ArovKrein problem. The rst one consists in extending a function given on a proper subset of the unit circle to the whole circle so as to make it as close as possible to meromorphic with prescribed number of poles, in the sup norm, while meeting some gauge constraint. The second consists in directly approximating the given function on the proper subset by the restriction of a meromorphic function, again meeting some gauge constraint. 1 Introduction 1.1 Framework and motivation The problem under study in this paper may be considered as a means of making the analytic continuation principle computationally eoeective in the disk for a function which is merely known on a subset of the circle. This issue arises in a variety of inverse problems ranging from deconvolution and the identication of linear control systems or stochastic processes to inverse scattering and NeumannDirichlet problems, see e.g. [6, 10, 13, 1..