Skip to Main Content

Mathematics of Computation

Published by the American Mathematical Society since 1960 (published as Mathematical Tables and other Aids to Computation 1943-1959), Mathematics of Computation is devoted to research articles of the highest quality in computational mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.78.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.


Refinable bivariate quartic $C^2$-splines for multi-level data representation and surface display
HTML articles powered by AMS MathViewer

by Charles K. Chui and Qingtang Jiang PDF
Math. Comp. 74 (2005), 1369-1390 Request permission


In this paper, a second-order Hermite basis of the space of $C^2$-quartic splines on the six-directional mesh is constructed and the refinable mask of the basis functions is derived. In addition, the extra parameters of this basis are modified to extend the Hermite interpolating property at the integer lattices by including Lagrange interpolation at the half integers as well. We also formulate a compactly supported super function in terms of the basis functions to facilitate the construction of quasi-interpolants to achieve the highest (i.e., fifth) order of approximation in an efficient way. Due to the small (minimum) support of the basis functions, the refinable mask immediately yields (up to) four-point matrix-valued coefficient stencils of a vector subdivision scheme for efficient display of $C^2$-quartic spline surfaces. Finally, this vector subdivision approach is further modified to reduce the size of the coefficient stencils to two-point templates while maintaining the second-order Hermite interpolating property.
  • Peter Alfeld and Larry L. Schumaker, Smooth macro-elements based on Powell-Sabin triangle splits, Adv. Comput. Math. 16 (2002), no. 1, 29–46. MR 1888218, DOI 10.1023/A:1014299228104
  • Charles K. Chui, Multivariate splines, CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 54, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1988. With an appendix by Harvey Diamond. MR 1033490, DOI 10.1137/1.9781611970173
  • Charles K. Chui, Vertex splines and their applications to interpolation of discrete data, Computation of curves and surfaces (Puerto de la Cruz, 1989) NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci., vol. 307, Kluwer Acad. Publ., Dordrecht, 1990, pp. 137–181. MR 1064960
  • C. K. Chui, H. C. Chui, and T. X. He, Shape-preserving interpolation by bivariate $C^1$ quadratic splines, Workshop on Computational Geometry (Torino, 1992) World Sci. Publ., River Edge, NJ, 1993, pp. 21–75. MR 1339308
  • Charles K. Chui and Qingtang Jiang, Surface subdivision schemes generated by refinable bivariate spline function vectors, Appl. Comput. Harmon. Anal. 15 (2003), no. 2, 147–162. MR 2007056, DOI 10.1016/S1063-5203(03)00062-9
  • N. Dyn, D. Levin, J. A. Gregory, A butterfly subdivision scheme for surface interpolation with tension control, ACM Trans. Graphics 2 (1990), 160–169.
  • Rong-Qing Jia, Shift-invariant spaces and linear operator equations, Israel J. Math. 103 (1998), 259–288. MR 1613580, DOI 10.1007/BF02762276
  • Rong-Qing Jia and Qing-Tang Jiang, Approximation power of refinable vectors of functions, Wavelet analysis and applications (Guangzhou, 1999) AMS/IP Stud. Adv. Math., vol. 25, Amer. Math. Soc., Providence, RI, 2002, pp. 155–178. MR 1887509, DOI 10.1090/amsip/025/13
  • Rong-Qing Jia and Qingtang Jiang, Spectral analysis of the transition operator and its applications to smoothness analysis of wavelets, SIAM J. Matrix Anal. Appl. 24 (2003), no. 4, 1071–1109. MR 2003322, DOI 10.1137/S0895479801397858
  • Rong Qing Jia and Charles A. Micchelli, On linear independence for integer translates of a finite number of functions, Proc. Edinburgh Math. Soc. (2) 36 (1993), no. 1, 69–85. MR 1200188, DOI 10.1017/S0013091500005903
  • L. Kobbelt, $\sqrt {3}$-subdivision, In Computer Graphics Proceedings, Annual Conference Series, 2000, pp. 103–112.
  • U. Labsik, G. Greiner, Interpolatory $\sqrt {3}$-subdivision, Proceedings of Eurographics 2000, Computer Graphics Forum, vol. 19, 2000, pp. 131–138.
  • C. Loop, Smooth subdivision surfaces based on triangles, Master’s thesis, University of Utah, Department of Mathematics, Salt Lake City, 1987.
  • G. Nürnberger and F. Zeilfelder, Developments in bivariate spline interpolation, J. Comput. Appl. Math. 121 (2000), no. 1-2, 125–152. Numerical analysis in the 20th century, Vol. I, Approximation theory. MR 1780046, DOI 10.1016/S0377-0427(00)00346-0
  • M. J. D. Powell and M. A. Sabin, Piecewise quadratic approximations on triangles, ACM Trans. Math. Software 3 (1977), no. 4, 316–325. MR 483304, DOI 10.1145/355759.355761
  • Paul Sablonnière, Error bounds for Hermite interpolation by quadratic splines on an $\alpha$-triangulation, IMA J. Numer. Anal. 7 (1987), no. 4, 495–508. MR 968521, DOI 10.1093/imanum/7.4.495
Similar Articles
  • Retrieve articles in Mathematics of Computation with MSC (2000): 65D07, 65D18, 41A15
  • Retrieve articles in all journals with MSC (2000): 65D07, 65D18, 41A15
Additional Information
  • Charles K. Chui
  • Affiliation: Department of Mathematics and Computer Science, University of Missouri–St. Louis, St. Louis, Missouri 63121 and Department of Statistics, Stanford University, Stanford, California 94305
  • Email:
  • Qingtang Jiang
  • Affiliation: Department of Mathematics and Computer Science, University of Missouri–St. Louis, St. Louis, Missouri 63121
  • Email:
  • Received by editor(s): July 8, 2003
  • Received by editor(s) in revised form: January 2, 2004
  • Published electronically: July 28, 2004
  • Additional Notes: The first author was supported in part by NSF Grants #CCR-9988289 and #CCR-0098331, and ARO Grant #DAAD 19-00-1-0512.
    The second author was supported in part by University of Missouri–St. Louis Research Award 03
  • © Copyright 2004 American Mathematical Society
  • Journal: Math. Comp. 74 (2005), 1369-1390
  • MSC (2000): Primary 65D07, 65D18; Secondary 41A15
  • DOI:
  • MathSciNet review: 2137007