A novel construction of B-spline-like bases for a family of many knot spline spaces and their application to quasi-interpolation

We define a family of univariate many knot spline spaces of arbitrary degree defined on an initial partition that is refined by adding a point in each sub-interval. For an arbitrary smoothness r, splines of degrees 2r and 2r + 1 are considered by imposing additional regularity when necessary. For an arbitrary degree, a B-spline-like basis is constructed by using the Bernstein-Bezier representation. Blossoming is then used to establish a Marsden's identity from which several quasi-interpolation operators having optimal approximation orders are defined. (c) 2021 Elsevier B.V. All rights reserved.

Automated summary

This article defines a family of univariate many knot spline spaces of arbitrary degree on an initial partition that is refined by adding a point in each sub-interval. Additional regularity is imposed when necessary for splines of degrees 2r and 2r + 1 with arbitrary smoothness r. A B-spline-like basis is constructed using the Bernstein-Bezier representation for arbitrary degree. Various quasi-interpolation operators with optimal approximation orders are defined by establishing a Marsden's identity through blossoming.