Comptes Rendus
Numerical Analysis/Mathematical Analysis
Chebyshevian splines: interpolation and blossoms
Comptes Rendus. Mathématique, Volume 344 (2007) no. 1, pp. 65-70.

We state and discuss a theorem which links the existence of blossoms in a spline space (with sections in different Extended Chebyshev spaces and with connection matrices which are not necessarily totally positive) with the possibility of Hermite interpolation in its derivative space under Schoenberg–Whitney conditions.

Cette note établit un lien fondamental entre existence de floraisons dans un espace de splines (à sections dans différents espaces de Chebyshev généralisés et avec matrices de connexion non nécessairement totalement positives) et possibilité d'interpoler au sens d'Hermite sous conditions de Schoenberg–Whitney.

Published online:
DOI: 10.1016/j.crma.2006.11.021
Alexander Kayumov 1; Marie-Laurence Mazure 2

1 Function Approximation Theory Department, Institute of Mathematics and Mechanics, 16, Sofia Kovalevskaya st., 620219 Ekaterinburg GSP-384, Russia
2 Laboratoire de modélisation et calcul (LMC-IMAG), université Joseph-Fourier, BP 53, 38041 Grenoble cedex 9, France
     author = {Alexander Kayumov and Marie-Laurence Mazure},
     title = {Chebyshevian splines: interpolation and blossoms},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {65--70},
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     year = {2007},
     doi = {10.1016/j.crma.2006.11.021},
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%A Marie-Laurence Mazure
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Alexander Kayumov; Marie-Laurence Mazure. Chebyshevian splines: interpolation and blossoms. Comptes Rendus. Mathématique, Volume 344 (2007) no. 1, pp. 65-70. doi : 10.1016/j.crma.2006.11.021.

[1] P.J. Barry De Boor-Fix dual functionals and algorithms for Tchebycheffian B-spline curves, Constructive Approximation, Volume 12 (1996), pp. 385-408

[2] C. de Boor; R. DeVore A geometric proof of total positivity for spline interpolation, Mathematics of Computation, Volume 45 (1985) no. 172, pp. 497-504

[3] M.-L. Mazure Blossoming: a geometrical approach, Constructive Approximation, Volume 15 (1999), pp. 33-68

[4] M.-L. Mazure Chebyshev splines beyond total positivity, Advances in Computational Mathematics, Volume 14 (2001), pp. 129-156

[5] M.-L. Mazure On the equivalence between existence of B-spline bases and existence of blossoms, Constructive Approximation, Volume 20 (2004), pp. 603-624

[6] M.-L. Mazure; H. Pottmann Tchebycheff curves (M. Gasca; C.A. Micchelli, eds.), Total Positivity and Its Applications, Kluwer, Dordrecht, 1996, pp. 187-218

[7] G. Mühlbach, One sided Hermite interpolation by piecewise different generalized polynomials, Advances in Computational Mathematics, in press

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