Spline quasi-interpolation in the Bernstein basis on the Powell–Sabin 6-split of a type-1 triangulation

In this paper, we provide quasi-interpolation schemes defined on a uniform triangulation of type-1 endowed with a Powell–Sabin refinement. In contrast to the usual construction of quasi interpolation splines on the 6-split, the approach described in this work does not require a set of appropriate basis functions. The approximating splines are directly defined by setting their Bézier ordinates to suitable combinations of the given data values. The resulting quasi-interpolants are C1 continuous and reproduce quadratic polynomials. Some numerical tests are given to confirm the theoretical results.