# Support and Approximation Properties of Hermite Splines

## TL;DR

This paper analyzes Hermite splines, showing they are optimally localized and have approximation capabilities comparable to cubic B-splines, making them highly suitable for computer graphics and geometric design.

## Contribution

It provides a formal investigation of Hermite splines' localization and approximation properties, highlighting their optimal support size and asymptotic similarity to cubic B-splines.

## Key findings

- Hermite splines have minimal support among functions with the same reproduction properties.
- They are asymptotically equivalent to cubic B-splines in approximation power.
- Hermite splines combine localization, approximation, and interpolation advantages.

## Abstract

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.