On a new criterion to decide whether a spline space can be used for design
Using the most suitable approach of blossoming, the author considers the question of usefulness of piecewise approximations taken from extended Chebyshev spaces for design purposes. The idea is to take at first piecewise functions of the spline type (classically these would be piecewise polynomial functions), each piece being from an extended (generalised) Chebyshev space instead of just a polynomial space of degree \(n\), say. Theses pieces are linked to each other under the conditions of smoothness of some order at the knots, where in this situation the smoothness is considered by conditions on the parameters. In this generality, it is not evident which are the cases where the resulting functions are suitable for design and which are not. It is the purpose of the paper to find well-applicable conditions for that question and this is nicely achieved by the tool of blossoming. The paper provides a good overview of the framework of the approach at the beginning. The criterion is provided in Section~3. Useful, illuminating examples are given for the case \(n=3\) (in the classical polynomial case that is the degree of the piecewise polynomials) and the special case of zero-multiplicity at the knots. That and some further remarks on some other important special cases end this interesting work.
scientific article; zbMATH DE number 6117474
scientific article