دانلود رایگان مقاله طراحی با فضاهای تکه ای چبیشف شبه تمدید شده

Abstract

A Quasi Extended Chebyshev (QEC) space is a space of sufficiently differentiable functions in which any Hermite interpolation problem which is not a Taylor problem is unisolvent. On a given interval the class of all spaces which contains constants and for which the space obtained by differentiation is a QEC-space has been identified as the largest class of spaces (under ordinary differentiability assumptions) which can be used for design. As a first step towards determining the largest class of splines for design, we consider a sequence of QEC-spaces on adjacent intervals, all of the same dimension, we join them via connection matrices, so as to maintain both the dimension and the unisolvence. The resulting space is called a Quasi Extended Chebyshev Piecewise (QECP) space. We show that all QECP-spaces are inverse images of two-dimensional Chebyshev spaces under piecewise generalised derivatives associated with systems of piecewise weight functions. We show illustrations proving that QECP-spaces can produce interesting shape effects.

8. Conclusion

In this article we have identified the class of all QECP-spaces. This is interesting from a theoretical side, but the illustrations provided show that their construction also permits interesting shape effects. In a QECP-space E assumed to be good for design, the presence of blossoms and their pseudoaffinity makes it possible to develop results for splines based on E, with any multiplicities at the interior knots, exactly as we did for splines based on one single QEC-space in Mazure (2011d), or earlier for splines based on a given ECP-space in Mazure (1999). As usual, the forcefulness of blossoms is that their pseudoaffinity guarantees that such spline spaces satisfy all expected properties for design (development of all the classical CAGD algorithms for splines; presence of quasi-B-spline bases which are the optimal normalised totally positive bases). This is thus one step towards the identification of the largest possible class of splines (with ordinary differentiability assumptions on the sections) which can be used not only for Design, but also for Approximation, Interpolation, and even Isogeometric Analysis. The complete identification of this class, announced in Mazure (2015), will be the object of a further article.