Local Explicit Many-Knot Spline Hermite Approximation Schemes.

Compactly supported fundamental functions for spline interpolation

“…After commenting on the exactness of the Catmull Rom procedure which appears in the context of Computer Aided Geometric Design [1] we generalize the results in [7] on what is referred to there as many knot spline interpolation. In particular, our derivation of optimal degree of exactness is totally different from that in [7]. Section 4 will be devoted to arbitrary knot sequences X and any order splines.…”

“…The basic idea for getting around this undesirable feature is to interpolate by splines with knots at T = {ti}~zcIR, where X is strictly contained in T. The additional knots in T \ X increase the flexibility of the splines and can be used to construct compactly supported fundamental splines Li. This idea has already been considered by others [7,8] and even has been applied to grid generation, 1' -42.…”

“…Some special cases covered by the present approach have appeared in the literature 1-1, 4, 7, 8] mainly, however, for the case of equidistant knots (cf. [7,8]) X = Z where in particular [7] has served as a major source of motivation for us. Of course, the cases studied in this paper despite their generality, still leave further possibilities for the selection of the interpolant and it would certainly be interesting to consider variations of this problem.…”

“…In the case of equidistant interpolation points, that is, X=TZ examples of local interpolation schemes have been given in [1,7,8]. In this section we will comment on a version of the Catmull Rom method (cf.…”

Spline Interpolation and Wavelet Construction

“…The fundamental spline function L n ,0 , for the case M Å 0, has been constructed by Qi [21]. Qi's result has been extended by Dahmen et al [7] to splines with nonuniform knots.…”