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Existence of setinterpolating and energyminimizing curves
 Comput. Aided Geom. Design
, 2003
"... Abstract. We consider existence of curves c: [0, 1] → R n which minimize an energy of the form ∫ �c (k) � p (k = 1, 2,..., 1 < p < ∞) under sideconditions of the form Gj(c(t1,j),..., c (k−1) (tk,j)) ∈ Mj, where Gj is a continuous function, ti,j ∈ [0, 1], Mj is some closed set, and the indi ..."
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Abstract. We consider existence of curves c: [0, 1] → R n which minimize an energy of the form ∫ �c (k) � p (k = 1, 2,..., 1 < p < ∞) under sideconditions of the form Gj(c(t1,j),..., c (k−1) (tk,j)) ∈ Mj, where Gj is a continuous function, ti,j ∈ [0, 1], Mj is some closed set, and the indices j range in some index set J. This includes the problem of finding energy minimizing interpolants restricted to surfaces, and also variational nearinterpolating problems. The norm used for vectors does not have to be Euclidean. It is shown that such an energy minimizer exists if there exists a curve satisfying the side conditions at all, and if among the interpolation conditions there are at least k points to be interpolated. In the case k = 1, some relations to arc length are shown. 1.
Variational interpolation of subsets
 Constr. Approx
, 2004
"... Abstract. We consider the problem of variational interpolation of subsets of Euclidean spaces by curves such that the L 2 norm of the second derivative is minimized. It is well known that the resulting curves are cubic spline curves. We study geometric boundary conditions arising for various types o ..."
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Abstract. We consider the problem of variational interpolation of subsets of Euclidean spaces by curves such that the L 2 norm of the second derivative is minimized. It is well known that the resulting curves are cubic spline curves. We study geometric boundary conditions arising for various types of subsets such as subspaces, polyhedra, and submanifolds, and we indicate how solutions can be computed in the case of convex polyhedra. 1. Introduction and
An abstract formulation of variational refinement
 J. Approx. Theory
"... 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 IR d. The analysis is relevant to each level of refinement (the limit curves are not considered here). The curves are characterized by an application of ..."
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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 IR d. 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 ∗ T f + ˜ Λ ∗ w Γ(Λf) = 0. Existence and uniqueness are established under certain conditions. The problem here is a generalization of that studied in [19] 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.