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≤∞ if |f (j−1) (t i )−z ij |≤ɛ 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.