Piecewise Extended Chebyshev Spaces: a numerical test for design

Given a number of Extended Chebyshev (EC) spaces on adjacent intervals, all of the same dimension, we join them via convenient connection matrices without increasing the dimension. The global space is called a Piecewise Extended Chebyshev (PEC) Space. In such a space one can count the total number of zeroes of any non-zero element, exactly as in each EC-section-space. When this number is bounded above in the global space the same way as in its section-spaces, we say that it is an Extended Chebyshev Piecewise (ECP) space. A thorough study of ECP-spaces has been developed in the last two decades in relation to blossoms, with a view to design. In particular, extending a classical procedure for EC-spaces, ECP-spaces were recently proved to all be obtained by means of piecewise generalised derivatives. This yields an interesting constructive characterisation of ECP-spaces. Unfortunately, except for low dimensions and for very few adjacent intervals, this characterisation proved to be rather difficult to handle in practice. To try to overcome this difficulty, in the present article we show how to reinterpret the constructive characterisation as a theoretical procedure to determine whether or not a given PEC-space is an ECP-space. This procedure is then translated into a numerical test, whose usefulness is illustrated by relevant examples

Polynomial spaces revisited via weight functions

167-198International audienceExtended Chebyshev spaces are natural generalisations of polynomial spaces due to the same upper bounds on the number of zeroes. In a natural approach, many results of the polynomial framework have been generalised to the larger Chebyshevian framework, concerning Approximation Theory as well as Geometric Design. In the present work, we go the reverse way: considering polynomial spaces as examples of Extended Chebyshev spaces, we apply to them results specifically developed in the Chebyshevian framework. On a closed bounded interval, each Extended Chebyshev space can be defined by means of sequences of generalised derivatives which play the same rôle as the ordinary derivatives for polynomials. We recently achieved an exhaustive description of the infinitely many such sequences. Surprisingly, this issue is closely related to the question of building positive linear operators of the Bernstein type. As Extended Chebyshev spaces, one can thus search for all generalised derivatives which can be associated with polynomials spaces on closed bounded intervals. Though this may a priori seem somewhat nonsensical due to the simplicity of the ordinary derivatives, this actually leads to new interesting results on polynomial and rational Bernstein operators and related results of convergence

Which spline spaces for design?

International audienceWe recently determined the largest class of spaces of sufficient regularity which are suitable for design. How can weconnect different such spaces, possibly with the help of connection matrices, to produce the largest class of splinesusable for design? We present the answer to this question, along with some of the major difficulties encountered toestablish it.We would like to stress that the results we announce are far from being a straightforward generalisationof previous work on piecewise Chebyshevian splines

A practical method for computing with piecewise Chebyshevian splines

A piecewise Chebyshevian spline space is good for design when it possesses a B-spline basis and this property is preserved under knot insertion. The interest in such kind of spaces is justified by the fact that, similarly as for polynomial splines, the related parametric curves exhibit the desired properties of convex hull inclusion, variation diminution and intuitive relation between the curve shape and the location of the control points. For a good-for-design space, in this paper we construct a set of functions, called transition functions, which allow for efficient computation of the B-spline basis, even in the case of nonuniform and multiple knots. Moreover, we show how the spline coefficients of the representations associated with a refined knot partition and with a raised order can conveniently be expressed by means of transition functions. This result allows us to provide effective procedures that generalize the classical knot insertion and degree raising algorithms for polynomial splines. We further discuss how the approach can straightforwardly be generalized to deal with geometrically continuous piecewise Chebyshevian splines as well as with splines having section spaces of different dimensions. From a numerical point of view, we show that the proposed evaluation method is easier to implement and has higher accuracy than other existing algorithms