*Article* **Scattered Data Interpolation and Approximation with Truncated Exponential Radial Basis Function**

#### **Qiuyan Xu and Zhiyong Liu \***

School of Mathematics and Statistics, Ningxia University, Yinchuan 750021, China; qiuyanxu@nxu.edu.cn **\*** Correspondence: zhiyong@nxu.edu.cn

Received: 14 October 2019; Accepted: 11 November 2019; Published: 14 November 2019

**Abstract:** Surface modeling is closely related to interpolation and approximation by using level set methods, radial basis functions methods, and moving least squares methods. Although radial basis functions with global support have a very good approximation effect, this is often accompanied by an ill-conditioned algebraic system. The exceedingly large condition number of the discrete matrix makes the numerical calculation time consuming. The paper introduces a truncated exponential function, which is radial on arbitrary *n*-dimensional space R*n* and has compact support. The truncated exponential radial function is proven strictly positive definite on R*n* while internal parameter *l* satisfies *l* ≥ *n* 2 + 1. The error estimates for scattered data interpolation are obtained via the native space approach. To confirm the efficiency of the truncated exponential radial function approximation, the single level interpolation and multilevel interpolation are used for surface modeling, respectively.

**Keywords:** radial basis functions; native spaces; truncated function; interpolation; approximation; surface modeling
