Piecewise Chebyshevian splines: interpolation versus design
The author investigates the connections between Hermite interpolation and geometric design in spline spaces, inspired by an analogous result in the non-spline case of a general W-space on a non trivial interval $I$, i.e. a space of sufficiently differentiable functions where the Wronskian of any basis never vanishes on $I$. The most general spline Hermite interpolation problem is considered by introducing the W-splines, possibily taking into account the presence of connection matrices. Necessary conditions are provided for a given interpolation problem to be unisolvent, possibly splitting it into several simpler ones. Blossoms, generalised derivatives, Schoenberg-Whitney conditions are tools used in the proof of the main result of the paper that claims that (i) a piecewise W-spline space ${\mathbf S}$ good for interpolation is equivalent to (ii) the piecewise W-spline space ${\widehat{\mathbf S}}$ ``good for design'', obtained from the first one by continuous integration on $I$. Also, if (i) is satisfied, the space ${\widehat{\mathbf S}}$ is ``good for interpolation'' in turn. Such a problem is explored in details, by also proving that in a good for design piecewise W-spline space appropriate systems of weight functions can always be found so that, in terms of the associated generalised derivatives, all matrices involved in the connection conditions are identity matrices, the simplest possible totally positive matrices. Taking advantage from this facts, the unisolvence of a Hermite interpolation problem is characterised in terms of upper bounds on the number of zeros of splines in ${\mathbf S}$. Some examples are reported on Hermite interpolation with W-splines and on geometrically continuous cubic splines, showing explicit necessary and sufficient conditions for the spline space to be good for interpolation and for design. There are also comparisons that show the equivalence between these two facts and a specific interest in unisolvence of interpolation problems, with special focus on ``interpolation beyond design'', by using parameters for local shape preservation and tension effects.
scientific article; zbMATH DE number 6860409
scientific article