A parametric curve f ∈ L (m) 2 ( [a, b]:n and j =1:m, and a "best near-interpolant" if it also minimizes b a |f (m) | 2 . In this paper optimality conditions are derived for these best near-interpolants. Based on these conditions it is shown that the near-interpolants are actually smoothing splines with weights that appear as Lagrange multipliers corresponding to the constraints. The optimality conditions are applied to the computation of near-interpolants in the last sections of the paper. Classification (1991): 41A05, 41A15, 41A29
Mathematics Subject
Abstract. In the first part of this paper we apply a saddle point theorem from convex analysis to show that various constrained minimization problems are equivalent to the problem of smoothing by spline functions. In particular, we show that near-interpolants are smoothing splines with weights that arise as Lagrange multipliers corresponding to the constraints in the problem of near-interpolation. In the second part of this paper we apply certain fixed point iterations to compute these weights. A similar iteration is applied to the computation of the smoothing parameter in the problem of smoothing.
the existence of minimizers to the nonlinear problem of best "interpolation" by curves are extended to the problem of best "near-interpolation" by curves that meet arbitrary sets, such as closed balls (as in [S. Kersey, Best Near-Interpolation by Curves: Optimality Conditions, Technical Report 99-05, Center for the Mathematical Sciences, University of Wisconsin, Madison, WI, 1999]). The minimizers are spline curves with breakpoints at the data sites at which the curves meet the sets, and the nonlinearities arise as these data sites vary from curve to curve. The results here apply to Hermite-type interpolation conditions, with the possibility of repeated data sites.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.