The thesis covers two topics. First, the completeness question is adressed, i.e., whether the hierarchical spline basis generates the entire space of piecewise polynomials on the corresponding hierarchical mesh. We identify sufficient conditions for a positive answer. We construct the truncated decoupled splines, which allow to further relax the assumptions on the mesh. Second, we discuss low-rank interpolation. A simple algorithm to generate a spline surface of rank at most 2d interpolating four boundary spline curves in R^d is presented and analyzed. Furthermore, for d = 2 a modification through a standardisation procedure is proposed in order to restore affine invariance of the algorithm. The algorithm is generalized for d = 1 to interpolate n cross-sections (not necessarily splines) in each direction with a surface of rank not exceeding n and its convergence rate is then shown to be equal to 2n.