Extended Chebyshev Spaces Research Articles

Towards existence of piecewise Chebyshevian B-spline bases

The paper addresses the problem of how to ensure existence of blossoms in the context of piecewise spaces built from joining different extended Chebyshev spaces by means of connection matrices. The interest of this issue lies in the fact that existence of blossoms is equivalent to existence of B-spline bases in all associated spline spaces. As is now classical, blossoms are defined in a geometric way by means of intersections of osculating flats. In such a piecewise context, intersecting a number of osculating flats is a tough proposition. In the present paper, we show that blossoms exist if an only if Bezier points exist, which significantly simplifies the problem. Existence of blossoms also proves to be equivalent to existence of Bernstein bases. In order to establish the latter results, we start by extending to the piecewise context some results which are classical for extended Chebyshev spaces.

Chebyshev Spaces and Bernstein Bases

We characterize extended Chebyshev spaces by the fact that any Hermite interpolation problem involving at most two different points has a unique solution. This enables us to prove that, in a given space, Bernstein bases exist if and only if the space obtained by differentiation is an extended Chebyshev space.

Blossoms and Optimal Bases

It is now classical to define blossoms by means of intersections of osculating flats. We consider here the most general context of spline spaces with sections in arbitrary extended Chebyshev spaces and with connections defined by arbitrary lower triangular matrices with positive diagonal elements. We show how the existence of blossoms in such spaces automatically leads to optimal bases in the sense of Carnicer and Pena.

On the Equivalence Between Existence of B-Spline Bases and Existence of Blossoms

In spline spaces with sections in arbitrary extended Chebyshev spaces and with connections defined by arbitrary lower triangular matrices with positive diagonal elements, we prove that existence of B-spline bases is equivalent to existence of blossoms. As is now classical, we construct blossoms with the help of osculating flats. As for B-spline bases, this expression denotes normalized basis consisting of minimally supported functions which are positive on the interior of their supports and which satisfy an additional “end point condition.”

Critical Length for Design Purposes and Extended Chebyshev Spaces

We analyze the connection between two ideas of apparently different nature. On one hand, the existence of an extended Chebyshev basis, which means that the Hermite interpolation problem has always a unique solution. On the other hand, the existence of a normalized totally positive basis, which means that the space is suitable for design purposes. We prove that the intervals where the existence of a normalized totally positive basis is guaranteed are those intervals where the existence of an extended Chebyshev basis of the space of derivatives can be ensured. We apply our results to the spaces Cn generated by 1,t, …, tn-2, cos t, sin t. In particular, C5 is a space suitable for design which permits the exact reproduction of remarkable parametric curves, including lines and circles with a single control polygon. We prove that this space has the minimal dimension for this purpose.

Blossoming beyond Extended Chebyshev Spaces

In a previous series of papers a theory of blossoming was developed for spaces of functions on an interval I spanned by the constant functions and functions Φ 1 , …, Φ n , where Φ ′ 1 , …, Φ ′ n span an extended Chebyshev space. This theory was then used to construct a generalisation of the Bernstein basis and the de Casteljau algorithm. Also considered were functions defined to be piecewise in such spaces, leading to generalisations of B-splines and the de Boor algorithm. Here we relax the condition that Φ ′ 1 , …, Φ ′ n span an extended Chebyshev space, while retaining all the nice properties of the earlier theory. This allows us to include a large variety of new spaces, including spaces of polynomials which have been found to be successful for tension methods for shape-preserving interpolation.

Chebyshev splines beyond total positivity

For polynomial splines as well as for Chebyshev splines, it is known that total positivity of the connection matrices is sufficient to obtain B-spline bases. In this paper we give a necessary and sufficient condition for the existence of B-bases (or, equivalently, of blossoms) for splines with connection matrices and with sections in different four-dimensional extended Chebyshev spaces.

Chebyshev–Bernstein bases

From a design point of view, extended Chebyshev spaces are interesting in so far as they provide useful shape parameters. In such spaces, the blossoming principle generates de Casteljau type algorithms, which makes it natural to define Chebyshev–Bernstein bases as the dual bases of the linear functional giving the control points. Such bases share the same properties as the Bernstein bases in polynomial spaces. In particular they are the optimal shape preserving bases. This also holds in piecewise smooth extended Chebyshev spaces.

Chebyshev spaces with polynomial blossoms

The use of extended Chebyshev spaces in geometric design is motivated by the interesting shape parameters they provide. Unfortunately the algorithms such spaces lead to are generally complicated because the blossoms themselves are complicated. In order to make up for this inconvenience, we here investigate particular extended Chebyshev spaces, containing the constants and power functions whose exponents are consecutive positive integers. We show that these spaces lead to simple algorithms due to the fact that the blossoms are polynomial functions. Furthermore, we also describe an elegant dimension elevation algorithm which makes it possible to return to polynomial spaces and therefore to use all the classical algorithms for polynomials.

Bernstein bases in Müntz spaces

Extended Chebyshev spaces possess Bernstein type bases. In this paper, we determine the expressions of such bases in spaces spanned by the constants and power functions with consecutive integer exponents.

Vandermonde type determinants and blossoming

For extended Chebyshev spaces spanned by power functions, the blossoms can be expressed by means of Vandermonde type determinants. When the exponents are nonnegative integers, it is possible to use the classical algorithms for polynomial functions after one or several dimension elevation processes. This provides interesting shape parameters.