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Optimal quasi-interpolation by quadratic $$C^1$$-splines on type-2 triangulations. (English) Zbl 1074.65015
Chui, Charles K.(ed.) et al., Approximation theory XI. Proceedings of the 11th international conference, Gatlinburg, TN, USA, May 18–22, 2004. Brentwood, TN: Nashboro Press (ISBN 0-9728482-5-8/hbk). Modern Methods in Mathematics, 423-438 (2005).
Summary: We describe a new scheme based on quadratic $$C^1$$-splines on type-2 triangulations, approximating gridded 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. In this way, 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.
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 optimal approximation order for smooth functions and we provide explicit constants in the corresponding error bounds.
For the entire collection see [Zbl 1061.41001].

##### MSC:
 65D05 Numerical interpolation 65D07 Numerical computation using splines 41A05 Interpolation in approximation theory 41A15 Spline approximation