The most essential ingredient of interpolation is its basis function. We have shown in previous papers that this basis need not be necessarily interpolating to achieve good results. On the contrary, several recent studies have confirmed that non-interpolating bases, such as B-splines and O-moms, perform best. This opens up a much wider choice of basis functions. In this paper, we give to the designer the tools that will allow him to characterize this enlarged space of functions. In particular, he will be able to specify up-front the four most important parameters for image processing: degree, support, regularity, and order. The theorems presented here will then allow him to refine his design by dealing with additional coefficients that can be selected freely, without interfering with the main design parameters.