Sum of Squares Decompositions in H\"older Spaces

TL;DR

This paper studies how many squares are needed to decompose non-negative functions with certain smoothness in H"older spaces, refining techniques and providing bounds for SOS decompositions.

Contribution

It introduces new bounds and unifies existing results on sum of squares decompositions for non-negative functions in H"older spaces.

Findings

Established upper and lower bounds on the number of squares needed

Unified various known SOS decomposition results

Refined techniques for SOS 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 $C^{k,\alpha}(\mathbb{R}^n)$.