Best Constrained Approximation in Hilbert Space and Interpolation by Cubic Splines Subject to Obstacles

We review a Lagrangian parameter approach to problems of best constrained approximation in Hilbert space. The variable is confined to a closed convex subset of the Hilbert space and is also assumed to satisfy linear equalities. The technique is applied to the problem of interpolation of data in a plane by a cubic spline function which is subject to obstacles. The obstacles may be piecewise cubic polynomials over the original knot set. A characterization result is obtained which is used to develop a Newton-type algorithm for the numerical solution.