L1-regularization methods have shown to be very powerful for a large variety of image processing modalities as they tend to promote sparse solutions. From a theoretical point of view, these methods exhibit strong links with the spline theory, in which a signal is described as a piecewise smooth function generated by a differential operator. This connection is well-understood for instance for the simple case of piecewise constant functions, which correspond to the derivative operator. The goal of this project is to develop a theory for a more general class of operators. It will involve studying the conditions of invertibility of differential operators, with a particular emphasis on fractional operators. The student should show a strong interest for the mathematical foundations of image processing, as well as good mathematical capabilities.