Pseudoaffinity, de Boor algorithm, and blossoms
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Springer 2006
Pseudoaffinity, de Boor algorithm, and blossoms Marie-Laurence Mazure Laboratoire de Modélisation et Calcul (LMC-IMAG), Université Joseph Fourier, BP 53, 38041 Grenoble Cedex, France E-mail: [email protected]
Received 15 December 2003; accepted 13 December 2004 Communicated by J. Carnicer and J.M.Peña
I would like to make a present of this article to Mariano Gasca to celebrate many fruitful years of research and above all this, his 60th birthday. I would also like to thank him for his permanent cheerfulness and simplicity of nature which have made each encounter with him a pleasure to be looked forward to.
In order to ensure existence of a de Boor algorithm (hence of a B-spline basis) in a given spline space with (n + 1)-dimensional sections, it is important to be able to generate each spline by restriction to the diagonal of a symmetric function of n variables supposed to be pseudoaffine w.r. to each variable. We proved that a way to obtain these three properties (symmetry, n-pseudoaffinity, diagonal property) is to suppose the existence of blossoms on the set of admissible n-tuples, given that blossoms are defined in a geometric way by means of intersections of osculating flats. In the present paper, we examine the converse: do symmetry, n-pseudoaffinity, and diagonal property imply existence of blossoms? Keywords: B-spline bases, blossoming, Chebyshev spaces, Chebyshev splines, geometric design Mathematics subject classification (2000): 65D17
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Introduction
In any spline space, a B-spline basis is usually meant as a normalized basis composed of minimally supported functions positive on the interior of their supports. Their existence, of fundamental importance in geometric design, is intimately connected with the existence of a de Boor algorithm. Consider first a space S of polynomial splines of degree n, with, at each knot tk , parametric continuity conditions of some order rk . In this situation, it is easy to point out the existence of a B-spline basis. Indeed it is a direct emanation from the de Boor algorithm, and since Ramshaw [13,14], we know how simple the description of the de Boor algorithm is when using blossoms. This is due to the three well-known properties
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M.-L. Mazure / Pseudoaffinity, de Boor algorithm, and blossoms
of polynomial blossoms: symmetry, n-affinity (i.e., affinity in each variable), diagonal property. Although classical, let us state this in detail in order to clarify the purpose of the present paper. With any polynomial P of degree at most n, we can associate the unique function p of n variables, called the blossom of P , which is symmetric on Rn , n-affine, and which gives P by restriction to the diagonal of Rn . It is well known that all geometric design algorithms for polynomials have an elegant description through blossoms. On the other hand, the fact that polynomial blossoms are good tools to express parametric contact conditions between two polynomials, makes it possible to associate with each spline S ∈ S a function s of n variables possessing the sam
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