Near-Interpolation

A parametric curve f ∈ L₂^(m) ([a, b] → ℝ^d) is a "near-interpolant" to prescribed data z _ij ∈ ℝ^d at data sites t_i ∈ [a, b] within tolerances 0<ε_ij≤∞ if |f ^(j-1)(t_i) - z_ij| ≤ ε_ij for i=1:n and j=1:m, and a "best near-interpolant" if it also minimizes ∫_a^b |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.