Rational Quasi-Interpolation Approximation of Scattered Data in $\mathbb{R}^3$

Abstract

This paper is concerned with a piecewise smooth rational quasi-interpolation with algebraic accuracy of degree $(n+1)$ to approximate the scattered data in $\mathbb{R}^3$. We firstly use the modified Taylor expansion to expand the mean value coordinates interpolation with algebraic accuracy of degree one to one with algebraic accuracy of degree $(n+1)$. Then, based on the triangulation of the scattered nodes in $\mathbb{R}^2$, on each triangle a rational quasi-interpolation function is constructed. The constructed rational quasi-interpolation is a linear combination of three different expanded mean value coordinates interpolations and it has algebraic accuracy of degree $(n+1)$. By comparing accuracy, stability, and efficiency with the $C^1$-Tri-interpolation method of Goodman [16] and the MQ Shepard method, it is observed that our method has some computational advantages.