A few remarks on recurrence relations for geometrically continuous piecewise Chebyshevian B-splines |
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Authors: | Marie-Laurence Mazure |
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Affiliation: | (1) Laboratoire Jean Kuntzmann, Université Joseph Fourier, BP53, 38041 Grenoble Cedex 9, France |
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Abstract: | This works complements a recent article (Mazure, J. Comp. Appl. Math. 219(2):457–470, 2008) in which we showed that T. Lyche’s recurrence relations for Chebyshevian B-splines (Lyche, Constr. Approx. 1:155–178, 1985) naturally emerged from blossoms and their properties via de Boor type algorithms. Based on Chebyshevian divided differences,
T. Lyche’s approach concerned splines with all sections in the same Chebyshev space and with ordinary connections at the knots.
Here, we consider geometrically continuous piecewise Chebyshevian splines, namely, splines with sections in different Chebyshev
spaces, and with geometric connections at the knots. In this general framework, we proved in (Mazure, Constr. Approx. 20:603–624,
2004) that existence of B-spline bases could not be separated from existence of blossoms. Actually, the present paper enhances
the powerfulness of blossoms in which not only B-splines are inherent, but also their recurrence relations. We compare this
fact with the work by G. Mühlbach and Y. Tang (Mühlbach and Tang, Num. Alg. 41:35–78, 2006) who obtained the same recurrence relations via generalised Chebyshevian divided differences, but only under some total positivity
assumption on the connexion matrices. We illustrate this comparison with splines with four-dimensional sections. The general
situation addressed here also enhances the differences of behaviour between B-splines and the functions of smaller and smaller
supports involved in the recurrence relations. |
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Keywords: | Chebyshev spaces Recurrence relations B-splines Connexion matrices Blossoms de Boor algorithm Geometric design |
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