L Infinity-Lower Bound of L2-Projections onto Splines on a Geometric Mesh

摘要

Least-squares approximation by polynomial splines is a very effective means of approximation, particularly when the knots are appropriately nonuniformly spaced to adapt to the particular behavior of the function being approximated. Unfortunately, the stability of this process has been established only for nearly uniform knot sequences. The stability can be linked to the norm of the inverse of the Gram matrix of a (appropriately scaled) B-spline basis. In an earlier report, we studied an important special case, that of a geometric knot sequence and there showed the norm of the inverse of that Gramian to be bounded independent of the mesh ratio. In the present report, we continue these investigations and show, in particular, the surprising fact that the norm of the inverse of the Gramian is least (i.e., the stability is greatest) when the mesh is most nonuniform. (Author)