A practical criterion for the existence of optimal piecewise Chebyshevian spline bases
For researchers in geometric design and spline theory, this offers a practical tool to verify design suitability of such spline spaces, which was previously lacking.
The paper provides a practical criterion and numerical procedure to determine whether a piecewise Chebyshevian spline space with zero-multiplicity knots is suitable for design (i.e., has an optimal basis preserved under knot insertion), and shows how to construct the optimal basis when it exists.
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.