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

## TL;DR

This paper improves the Finiteness Principle for nonnegative $ C^2(R^2) $ interpolation, reducing the finiteness constant and enhancing computational practicality, while also exploring one-dimensional interpolants and extension operators.

## Contribution

It presents two sharper versions of the Finiteness Principle for nonnegative $ C^2 $ interpolation and analyzes extension operators, advancing theoretical understanding and computational methods.

## Key findings

- Finiteness constant improved to 64
- Provided detailed construction of nonnegative $ C^2 $ interpolants in 1D
- Proved nonexistence of a bounded linear $ C^2 $ extension operator that preserves nonnegativity

## Abstract

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.