A functional generalization of the interpolation problem

Given two linearly independent functions f1 and f2, we generalize the interpolating problem to the space πn(f1,f2) spanned by the basis f1n-kf2kk=0n. We show that this problem has a unique solution and represent this solution by a functional analogue of the Lagrange formula. We also give a similar generalization of Hermite interpolation.