For the robust $H^\infty$ identification of linear shift-invariant systems from frequency responses, the so-called two-stage algorithms are widely used. If the second step of such algorithms is generally reduced to the resolution of a nonlinear Nehari extension problem, the first stage may lead to specific interpolation or approximation techniques depending on the nature of the available data (density, corruptness, distribution, \ldots). In this paper we present two approximation techniques. In section \ref{RobPolApp}, we study on a subset of the unit circle, the robust polynomial approximation based on a least-deviation problem. In section \ref{secJdlVP}, we generalize Partington's results about robust Jackson and de la Vallée Poussin polynomial approximation, to the case of non-equally spaced points densely distributed on a subset of the unite circle.