Multivariate normalized Powell–Sabin B-splines and quasi-interpolants

We present the construction of a multivariate normalized B-spline basis for the quadratic C1C1-continuous spline space defined over a triangulation in RsRs (s⩾1s⩾1) with a generalized Powell–Sabin refinement. The basis functions have a local support, they are nonnegative, and they form a partition of unity. The construction can be interpreted geometrically as the determination of a set of s-simplices that must contain a specific set of points. We also propose a family of quasi-interpolants based on this multivariate Powell–Sabin B-spline representation. Their spline coefficients only depend on a set of local function values. The multivariate quasi-interpolants reproduce quadratic polynomials and have an optimal approximation order.

► We construct a normalized basis for multivariate quadratic Powell–Sabin splines. ► A geometric interpretation is given in terms of simplices containing certain points. ► We also discuss a family of multivariate quasi-interpolants.