An explicit quasi-interpolation scheme based on C1 quartic splines on type-1 triangulations

We describe a new scheme based on quartic C1-splines on type-1 triangulations approximating regularly distributed data. The quasi-interpolating splines are directly determined by setting the Bernstein–Bézier coefficients of the splines to appropriate combinations of the given data values. Each polynomial piece of the approximating spline is immediately available from local portions of the data, without using prescribed derivatives at any point of the domain. Moreover, the operator interpolates the given data values at all the vertices of the underlying triangulation. Since the Bernstein–Bézier coefficients of the splines are computed directly, an intermediate step making use of certain locally supported splines spanning the space is not needed. We prove that the splines yield nearly-optimal approximation order for smooth functions. The order is known to be best possible for these spaces. Numerical tests confirm the theoretical behavior and show that the approach leads to functional surfaces of high visual quality.