Support and Approximation Properties of Hermite Splines
Provides theoretical justification for using Hermite splines in computer graphics and geometric design, though the results are incremental over known B-spline theory.
This paper proves that Hermite splines have minimal support among functions with the same reproduction properties and that their approximation power for functions and derivatives is asymptotically identical to cubic B-splines, combining optimal localization with strong approximation.
In this paper, we formally investigate two mathematical aspects of Hermite splines which translate to features that are relevant to their practical applications. We first demonstrate that Hermite splines are maximally localized in the sense that their support sizes are minimal among pairs of functions with identical reproduction properties. Then, we precisely quantify the approximation power of Hermite splines for reconstructing functions and their derivatives, and show that they are asymptotically identical to cubic B-splines for these tasks. Hermite splines therefore combine optimal localization and excellent approximation power, while retaining interpolation properties and closed-form expression, in contrast to existing similar approaches. These findings shed a new light on the convenience of Hermite splines for use in computer graphics and geometrical design.