Abstract
Let τ be the uniform triangulation generated by the usual three-directional mesh of the plane and let Σ1 be the unit square consisting of two triangles of τ. We study the space of piecewise polynomial functions in C k(R 2) with support Σ1 having a sufficiently high degree n, which are symmetrical with respect to the first diagonal of Σ1. Such splines are called Σ1-splines. We first compute the dimension of this space in function of n and k. Then, for any fixed k≥0, we prove the existence of Σ1-splines of class C k and minimal degree. These splines are not unique. Finally, we describe an algorithm computing the Bernstein–Bézier coefficients of these splines, and we give an example.
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References
P. Alfeld and L.L. Schumaker, Non-existence of star-supported spline bases, SIAM J. Math. Anal. 31 (2000) 1482–1501.
El. Ameur and D. Sbibih, Cardinal interpolation by H-splines on a three-directional mesh of the plane, submitted.
C.K. Chui, Multivariate Splines, CBMS-NSF Regional Conference Series in Applied Mathematics, Vol. 54 (SIAM, Philadelphia, PA, 1988).
C.K. Chui, Vertex Splines and their applications to interpolation of discrete data, in: Computation of Curves and Surfaces, eds. W. Dahmen et al. (Kluwer, Dordrecht, 1990) pp. 137–181.
C. de Boor, K. Höllig and S. Riemenschneider, Box-Splines, Applied Mathematical Sciences, Vol. 98 (Springer, New York, 1993).
X. Guillet, Interpolation by new families of B-splines on uniform meshes of the plane, in: Surface Fitting and Multiresolution Methods, eds. A. Le Méhauté, C. Rabut and L.L. Schumaker (Vanderbilt Univ. Press, Nashville/London, 1997) pp. 183–190.
A. Mazroui, P. Sablonnière and D. Sbibih, Existence and construction of H 1-splines of class C k on a three-direction mesh, Adv. Comput. Math. 17 (2002) 167–198.
A. Mazroui and S. Sbibih, Existence and construction of Δ1-splines of class C k on a three-directional mesh, Numer. Algorithms 26 (2001) 21–48.
A. Mazroui and D. Sbibih, Algorithms for B-nets of some B-splines with hexagonal support on a three-directional mesh, submitted.
P. Sablonnière, Quasi-interpolants associated with H-splines on a three-direction mesh, J. Comput. Appl. Math. 66 (1996) 433–442.
P. Sablonnière, New families of B-splines on uniform mesh of the plane, in: CRM Proceedings and Lecture Notes, Vol. 18, Centre de Recherche Mathématique de Montréal (1999) pp. 89–100.
P. Sablonnière and D. Sbibih, Some families of splines with hexagonal support on a three-direction mesh, in: Mathematical Methods for Curves and Surfaces, eds. M. Daehlen, T. Lyche and L.L. Schumaker (Vanderbilt Univ. Press, Nashville, 1995) pp. 467–475.
G. Strang and G. Fix, A Fourier analysis of the finite element variational method, in: Constructive Aspects of Functional Analysis, ed. G. Geymonat (CIME II Ciclo, 1971) pp. 793–840.
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Mazroui, A., Sbibih, D. & Tijini, A. C k-B-splines with Square Support on a Three-Direction Mesh of the Plane. Numerical Algorithms 34, 67–84 (2003). https://doi.org/10.1023/A:1026102126842
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DOI: https://doi.org/10.1023/A:1026102126842