In this paper, the theory of abstract splines is applied to the variational refinement of (periodic) curves that meet data to within convex sets in Rd. The analysis is relevant to each level of refinement (the limit curves are not considered here). The curves are characterized by an application of a separation theorem for multiple convex sets, and represented as the solution of an equation involving the dual of certain maps on an inner product space.

Namely, T*Tf+Λ˜*wΓ(Λf)=0. Existence and uniqueness are established under certain conditions. The problem here is a generalization of that studied in (Kersey, Near-interpolatory subdivided curves, author's home page, 2003) to include arbitrary quadratic minimizing functionals, placed in the setting of abstract spline theory. The theory is specialized to the discretized thin beam and interval tension problems.