In modern robust control, control synthesis may be cast as an interpolation problem where the interpolant relates to robustness and performance criteria. In particular, robustness in the gap fits into this framework and the magnitude of the corresponding interpolant dictate the robustness to perturbations of the plant as a function of frequency. In this paper we consider the correspondence between weighted entropy functionals and minimizing interpolants in order to find appropriate interpolants for e.g. control synthesis. There are two basic issues that we address: we first characterize admissible shapes of minimizers by studying the corresponding inverse problem, and then we develop effective ways of shaping minimizers via suitable choices of weights. These results are used in order to systematize feedback control synthesis to obtain frequency dependent robustness bounds with a constraint on the controller degree.