High order approximation to non-smooth multivariate functions

Approximations of non-smooth multivariate functions return low-order approximations in the vicinities of the singularities. Most prior works solve this problem for univariate functions. In this work we introduce a method for approximating non-smooth multivariate functions of the form $f = g + r_+$ where $g,r \in C^{M+1}(\mathbb{R}^n)$ and the function $r_+$ is defined by $$r_+(y) = \left\{ \begin{array}{ll} r(y), & r(y) \geq 0 \\ 0, & r(y) < 0 \end{array} \right. \ , \ \forall y \in \mathbb{R}^n \ .$$ Given scattered (or uniform) data points $X \subset \mathbb{R}^n$, we investigate approximation by quasi-interpolation. We design a correction term, such that the corrected approximation achieves full approximation order on the entire domain. We also show that the correction term is the solution to a Moving Least Squares (MLS) problem, and as such can both be easily computed and is smooth. Last, we prove that the suggested method includes a high-order approximation to the locations of the singularities.