Abstract
We study the following modification of a linear subdivision scheme S: let M be a surface embedded in Euclidean space, and P a smooth projection mapping onto M. Then the P-projection analogue of S is defined as T := P ◦ S. As it turns out, the smoothness of the scheme T is always at least as high as the smoothness of the underlying scheme S or the smoothness of P minus 1, whichever is lower. To prove this we use the method of proximity as introduced by Wallner et al. (Constr Approx 24(3):289–318, 2006; Comput Aided Geom Design 22(7):593–622, 2005). While smoothness equivalence results are already available for interpolatory schemes S, this is the first result that confirms smoothness equivalence properties of arbitrary order for general non-interpolatory schemes.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bump D.: Lie Groups. Springer, New York (2004)
Donoho, D.L.: Wavelet-type representation of Lie-valued data, 2001. Talk at the IMI “Approximation and Computation” meeting, May 12–17, 2001. Charleston, South Carolina
Dyn N.: Subdivision schemes in computer-aided geometric design. In: Light, W.A. (eds) Advances in Numerical Analysis, vol. II, pp. 36–104. Oxford University Press, Oxford (1992)
Grohs, P.: Smoothness of multivariate interpolatory subdivision in Lie groups. IMA J. Numer. Anal. (2009, to appear)
Grohs, P., Wallner, J.: Interpolatory wavelets for manifold-valued data. ACHA. (2009, to appear). http://www.geometrie.tugraz.at/wallner/wav.pdf
Grohs P., Wallner J.: Log-exponential analogues of univariate subdivision schemes in Lie groups and their smoothness properties. In: Neamtu, M., Schumaker, L.L. (eds) Approximation Theory XII: San Antonio 2007, pp. 181–190. Nashboro Press, Brentwood (2008)
Higham N.: Matrix nearness problems and applications. In: Gover, M.J.C., Barnett, S. (eds) Applications of Matrix Theory, pp. 1–27. Oxford University Press, New York (1989)
Micchelli C.A., Cavaretta A., Dahmen W.: Stationary Subdivision. American Mathematical Society, Boston (1991)
Wallner J.: Smoothness analysis of subdivision schemes by proximity. Constr. Approx. 24(3), 289–318 (2006)
Wallner J., Dyn N.: Convergence and C 1 analysis of subdivision schemes on manifolds by proximity. Comput. Aided Geom. Des. 22(7), 593–622 (2005)
Wallner J., Nava Yazdani E., Grohs P.: Smoothness properties of Lie group subdivision schemes. Multiscale Model. Simul. 6, 493–505 (2007)
Wallner J., Pottmann H.: Intrinsic subdivision with smooth limits for graphics and animation. ACM Trans. Graph. 25(6), 356–374 (2006)
Wilf H.S.: Generating Functionology. Academic Press, London (1994)
Xie G., Yu T.P.-Y.: Smoothness equivalence properties of manifold-valued data subdivision schemes based on the projection approach. SIAM J. Numer. Anal. 45(3), 1220–1225 (2007)
Yu T.P.-Y.: Cutting corners on the sphere. In: Chen, G., Lai, M.-J. (eds) Wavelets and Splines: Athens 2005, pp. 496–506. Nasboro Press, Brentwood (2006)
Yu T.P.-Y.: How data dependent is a nonlinear subdivision scheme? A case study based on convexity preserving subdivision. SIAM J. Numer. Anal. 44(3), 936–948 (2006)
Yu, T.P.-Y.: Smoothness equivalence properties of interpolatory Lie group subdivision schemes. IMA J. Numer. Anal. (2009, to appear)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Grohs, P. Smoothness equivalence properties of univariate subdivision schemes and their projection analogues. Numer. Math. 113, 163–180 (2009). https://doi.org/10.1007/s00211-009-0231-9
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00211-009-0231-9