In this work we consider robust stabilization of uncertain dynamical systems and show that this can be achieved by solving a non-classically constrained analytic interpolation problem. In particular, this non-classical constraint confines the range of the interpolant, when evaluated on the imaginary axis, to a frequency-dependent set. By considering a sufficient condition for when this interpolation problem has a solution, we derive an approximate solution algorithm that can also be used for controller synthesis. The conservativeness of the method is reduced by introducing a shift, which can be tuned by the user. Finally, the theory is illustrated on a numerical example with a plant with uncertain gain, phase, and output delay.