Unicity of best mean approximation by second order splines with variable knots Academic Article uri icon

abstract

  • Let S2Ndenote the nonlinear manifold of second order splines defined on [0, 1] having at most N interior knots, counting multiplicities. We consider the question of unicity of best approximations to a function f by elements of S2N Approxima¬tion relative to the L2[ 0, 1] norm is treated first, with the results then extended to the best L1 and best one-sided L1 approximation problems. The conclusions in each case are essentially the same, and can be summarized as follows: A sufficiently smooth function f satisfying f" > 0 has a unique best approximant from S2N provided either log f" is concave, or N is sufficiently large, N ≥N0(f); for any N, there is a smooth function f, with f" > 0, having at least two best approximants. A principal tool in the analysis is the finite dimensional topological degree of a mapping. © 1978 American Mathematical Society.

author list (cited authors)

  • Barrow, D. L., Chui, C. K., Smith, P. W., & Ward, J. D.

publication date

  • January 1, 1978 11:11 AM