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We study a variant of the interpolation problem where the continuously defined solution is regularized by minimizing the L p -norm of its second-order derivative. For this continuous-domain problem, we propose an exact discretization scheme that restricts the search space to quadratic splines with knots on an uniform grid. This leads to a discrete finite-dimensional problem that is computationally tractable. Another benefit of our spline search space is that, when the grid is sufficiently fine, it contains functions that are arbitrarily close to the solutions of the underlying unrestricted problem. We implement an iteratively reweighted algorithm with a grid-refinement strategy that computes the solution within a prescribed accuracy. Finally, we present experimental results that illustrate characteristics, such as sparsity, of the L p -regularized interpolants.