EPFL | Biomedical Imaging Group

Support and Approximation Properties of Hermite Splines

J. Fageot, S. Aziznejad, M. Unser, V. Uhlmann

Journal of Computational and Applied Mathematics, vol. 368, paper no. 112503, 15 p., April 2020.

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.

@ARTICLE(http://bigwww.epfl.ch/publications/fageot2001.html,
AUTHOR="Fageot, J. and Aziznejad, S. and Unser, M. and Uhlmann, V.",
TITLE="Support and Approximation Properties of {H}ermite Splines",
JOURNAL="Journal of Computational and Applied Mathematics",
YEAR="2020",
volume="368",
number="",
pages="",
month="April",
note="paper no.\ 112503")

© 2020 The Authors. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists, or to reuse any copyrighted component of this work in other works must be obtained from The Authors. This material is presented to ensure timely dissemination of scholarly and technical work. Copyright and all rights therein are retained by authors or by other copyright holders. All persons copying this information are expected to adhere to the terms and constraints invoked by each author's copyright. In most cases, these works may not be reposted without the explicit permission of the copyright holder.