Best l₂ Spline-By-Spline Approximation

It has been shown (by Lutterkort, Peters and Reif) that the problem of best approximation of a polynomial by one of lower degree (a.k.a., degree reduction) in the L2(a, b) norm is equivalent to degree reduction in the l2 norm of the degree-raised coefficients in the Bernstein basis. Noting that we can generally get a better approximation from the space of piecewise polynomials, we are interested in the best approximation of a polynomial by a spline function of lower degree with interior knots, and more generally, the best approximation of a spline by another with different degree and knot sequence. In particular, we develop the theory of best spline-by-spline approximation in the l2 norm of the coefficients embedded in certain common minimal spline space, and observe that this problem is NOT the same as in the L2(a, b) norm, with some minor exceptions. Finally, noting that we can approximate better with variable knots, we investigate the problem of best free-knot spline-by-spline l2 approximation (which is therefore different than the L2(a, b) theory that has been well-studied in the literature).