Piecewise Chebyshev–Schoenberg operators: Shape preservation, approximation and space embedding

Abstract

By piecewise Chebyshevian splines we mean splines with sections in different Extended Chebyshev spaces of the same dimension, given that we allow connection matrices at the knots. As special instances we find classical Chebyshevian splines, L-splines (with sufficiently small knot spacing), geometrically continuous polynomial splines. In this difficult general context, existence of B-spline bases and preservation of that existence under knot insertion is guaranteed by the presence of blossoms. Roughly speaking, we then say that the piecewise Chebyshevian spline space is good for design. We show that any such spline space provides us with infinitely many operators of the Schoenberg-type, which are automatically shape preserving. We take advantage of a recently achieved constructive description of all piecewise Chebyshevian spline spaces good for design to develop further properties of their piecewise Chebyshev–Schoenberg operators. In particular we show that they are intimately connected with special linear piecewise differential operators associated with systems of piecewise weight functions, with respect to which the connection matrices are identity matrices. Thanks to the properties of blossoms, this general context enables us to achieve simple sufficient conditions for simultaneous approximation of a function and of its first derivative, understood in the sense of the latter piecewise differential operators. Finally, we consider the behaviour of piecewise Chebyshev–Schoenberg operators under space embedding.