Good approximation by splines with variable knots +

Here, t1, . . . , tN are the knots or joints of s ∈ S k π, and their multiplicity, i.e., equality among two or more of these, indicates reduced smoothness at that knot location in the usual way. Best approximation to f by elements of S π is quite well understood for a variety of norms since, after all, S π is a linear space. It seems practically more important and theoretically more interesting to investigate the approximation to f by splines of order k with N knots, i.e., by elements of S N := ∪πS k π