Abstract
In this paper, we formally investigate two mathematical aspects of Hermite splines that are relevant to practical applications. We first demonstrate that Hermite splines are maximally localized, in the sense that the size of their support is minimal among pairs of functions with identical reproduction properties. Then, we precisely quantify the approximation power of Hermite splines for the reconstruction of functions and their derivatives. It is known that the Hermite and B-spline approximation schemes have the same approximation order. More precisely, their approximation error vanishes as O(T4) when the step size T goes to zero. In this work, we show that they actually have the same asymptotic approximation error constants, too. Therefore, they have identical asymptotic approximation properties. Hermite splines combine optimal localization and excellent approximation power, while retaining interpolation properties and closed-form expression, in contrast to existing similar functions. These findings shed a new light on the convenience of Hermite splines in the context of computer graphics and geometrical design.
Highlights
In his seminal 1973 monograph on cardinal interpolation and spline functions [1], I.J
The joint interpolation properties of Hermite splines, which ensure that, at integer values, the interpolated function exactly matches the sequences of samples and derivatives samples that were used to build it; their smoothness properties [11], which guarantee low curvature of the interpolated curve under some mild conditions; and their statistical optimality for the reconstruction of second-order Brownian motion from direct and first derivative samples [14]
We formally demonstrate that Hermite splines have the minimal support property among basis functions generating cubic and quadratic splines
Summary
In his seminal 1973 monograph on cardinal interpolation and spline functions [1], I.J. The Hermite interpolation problem involves two sequences of discrete samples, which impose constraints on the resulting interpolated function and on its derivatives up to a given order. The practical value of Hermite splines in this context is to offer tangential control on the interpolated curve. This can be understood through their link with Bézier curves [7]. Hermite splines provides a suitable solution to a number of problems, whether with respect to simplicity of construction, efficiency, or convenience. This hands-on intuition can be translated to formal properties of Hermite splines and mathematically characterized. We investigate in this work the theoretical counterpart of two additional features that are observed to grant Hermite splines their practical usefulness
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