Nonnegative $C^2(\mathbb{R}^2)$ interpolation

In this paper, we prove two improved versions of the Finiteness Principle for nonnegative $C^2(\mathbb{R}^2)$ interpolation, previously proven by Fefferman, Israel, and Luli. The first version sharpens the finiteness constant to $64$, and the second version carries better computational practicality. Along the way, we also provide detailed construction of nonnegative $C^2$ interpolants in one-dimension, and prove the nonexistence of a bounded linear $C^2$-extension operator that preserves nonnegativity.