A practical criterion for the existence of optimal piecewise Chebyshevian spline bases

TL;DR

This paper presents a practical criterion and numerical method to determine the existence of an optimal basis in piecewise Chebyshevian spline spaces, facilitating their use in design applications.

Contribution

It introduces a new criterion and procedure to identify and construct optimal bases in Chebyshevian spline spaces with zero-multiplicity knots, ensuring their suitability for design.

Findings

Provides a practical criterion for optimal basis existence.

Develops an effective numerical procedure for construction.

Ensures properties are preserved under knot insertion.

Abstract

A piecewise Chebyshevian spline space is a space of spline functions having pieces in different Extended Chebyshev spaces and where the continuity conditions between adjacent spline segments are expressed by means of connection matrices. Any such space is suitable for design purposes when it possesses an optimal basis (i.e. a totally positive basis of minimally supported splines) and when this feature is preserved under knot insertion. Therefore, when any piecewise Chebyshevian spline space where all knots have zero multiplicity enjoys the aforementioned properties, then so does any spline space with knots of arbitrary multiplicity obtained from it. In this paper, we provide a practical criterion and an effective numerical procedure to determine whether or not a given piecewise Chebyshevian spline space with knots of zero multiplicity is suitable for design. Moreover, whenever it exists, we also show how to construct the optimal basis of the space.