We study continuous-domain linear inverse problems with generalized total-variation (gTV) regularization, expressed in terms of a regularization operator L. It has recently been proved that such inverse problems have sparse spline solutions, with fewer jumps than the number of measurements. Moreover, the type of spline solely depends on L (L-splines) and is independent of the measurements. The continuous-domain inverse problem can be recast in an exact way as a finite-dimensional problem by restricting the search space to splines with knots on a uniform finite grid. However, expressing the L-spline coefficients in the dictionary basis of the Green's function of L is ill-suited for practical problems due to its infinite support. Instead, we propose to formulate the problem in the B-spline dictionary basis, which leads to better-conditioned problems. As we make the grid finer, we show that a solution of the continuous-domain problem can be approached arbitrarily closely with functions of this search space. This result motivates our proposed multiresolution algorithm, which computes sparse solutions of our inverse problem. We demonstrate that this algorithm is computationally feasible for 1D signals when L is an ordinary differential operator.