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 Bezier 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.(c) 2022 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

Automated summary

In this paper, a quasi-interpolation scheme is proposed on a uniform triangulation of type-1 with a Powell-Sabin refinement. Unlike the traditional construction of quasi-interpolation splines on the 6-split, the method described in this work does not require a set of appropriate basis functions. The resulting approximating splines are directly defined by setting their Bezier ordinates to suitable combinations of the given data values. Numerical tests are conducted to confirm the theoretical results.