Sum of Squares Decompositions in H\"older Spaces

Abstract

We investigate the number of half-regular squares required to decompose a non-negative $C^{k,\alpha}(\mathbb{R}^n)$ function into a sum of squares. Each non-negative $C^{3,1}(\mathbb{R}^n)$ function is known to be a finite SOS in $C^{1,1}(\mathbb{R}^n)$, and similar regularity-preserving SOS decompositions have been studied by various authors. Our work refines existing techniques to unify and build upon several known decomposition results, and moreover we provide upper and lower estimates on the number of squares required for SOS decompositions in $C^{k,\alpha}(\mathbb{R}^n)$.Comment: Includes several condensed and improved results from arXiv:2303.0799