**6. Conclusions**

The truncated exponential radial function, which has compact support, was introduced in the paper. The strictly positive definiteness of TERBF was proven via the multiply monotonicity approach, and the interpolation error estimates were obtained via the native space approach. Moreover, the TERBF was applied to 2D/3D scattered data interpolation and surface modeling successfully.

However, we found that <sup>Φ</sup>(**x**)=(*e*1−*<sup>ε</sup>* **<sup>x</sup>** − <sup>1</sup>)*<sup>l</sup>*+ was only in *C*<sup>0</sup> space. In the error estimates in terms of the fill distance, the power of *h*X ,Ω was only 1/2. There are many possibilities for enhancement of TERBF approximation:

(1) We can construct new strictly positive definite radial functions with finite smoothness from the given <sup>Φ</sup>(**x**) by a "dimension-walk" technique.

(2) We can do in-depth analysis of the characterization of TERBF in terms of Fourier transforms established by Bochner and Schoenberg's theorems.

(3) TERBF can also be used for the numerical solution of partial differential equations. The convergence proof will depend on the approximation of TERBF trial spaces, the appropriate inverse inequality, and the sampling theorem.

**Author Contributions:** Conceptualization, Methodology and Writing–original draft preparation, Q.X.; Formal analysis and Writing—review and editing, Z.L.

**Funding:** The research of the first author was partially supported by the Natural Science Foundations of Ningxia Province (No. NZ2018AAC03026) and the Fourth Batch of the Ningxia Youth Talents Supporting Program (No. TJGC2019012). The research of the second author was partially supported by the Natural Science Foundations of China (No. 11501313), the Natural Science Foundations of Ningxia Province (No. 2019AAC02001), the Project funded by the China Postdoctoral Science Foundation (No. 2017M621343), and the Third Batch of the Ningxia Youth Talents Supporting Program (No. TJGC2018037).

**Acknowledgments:** The authors would like to thank the Editor and two unknown reviewers who made valuable comments on an earlier version of this paper. The authors used some Halton datasets and drew lessons from partial codes from Fasshauer's book [3]. We are grateful to [3] for its free CD, which contains many MATLAB codes.

**Conflicts of Interest:** The authors declare no conflict of interest.
