Title: Point and differential <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="mml137" display="inline" overflow="scroll" altimg="si137.gif"><mml:msup><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:math> quasi-interpolation on three direction meshes
Abstract: In this paper we construct and analyse C1 cubic and quartic quasi-interpolating splines on type-1 triangulations approximating regularly distributed data, without using minimal determining sets and without defining the approximating splines as linear combinations of compactly supported bivariate spanning functions. In particular, the C1 cubic splines are directly determined by setting their Bernstein–Bézier coefficients to appropriate combinations of the given data values without using prescribed derivatives at any point of the domain, in such a way that the C1-smoothness conditions are satisfied and approximation order three is guaranteed, for smooth functions. We also propose some numerical tests that confirm the theoretical results. Then, from the above C1 cubic splines we obtain C1 quartic splines exact on P3, achieving approximation order four. The associated differential quasi-interpolation operator involves the values of the first partial derivatives in its definition.