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In this letter, construction of analytic functions from evaluations of real or imaginary parts on finite subsets of the unit circle is studied. The points in the subsets are not necessarily uniformly spaced as in the most existing works. An algorithm exactly retrieving finite-dimensional systems from noise-free data is presented. This algorithm is based on a recent frequency-domain subspace algorithm to identify discrete-time power spectra from non-uniformly spaced measurements and inherits consistency and robustness properties of the latter with respect to inaccuracies in data.