  In the process of proving a sharpened form of G\r{a}rding's inequality,
Fefferman & Phong demonstrated that every non-negative function $f\in
C^{3,1}(\mathbb{R}^n)$ can be written as a finite sum of squares of functions
in $C^{1,1}(\mathbb{R}^n)$. In this thesis, we generalize their decomposition
result to show that if $f\in C^{k,\alpha}(\mathbb{R}^n)$ is non-negative for
$0\leq k\leq 3$ and $0<\alpha\leq 1$, then $f$ can be written as a finite sum
of squares of functions that are each `half' as regular as $f$, in the sense
that they belong to the H\"older space \[
  C^\frac{k+\alpha}{2}(\mathbb{R}^n)=\begin{cases}\hfil
C^{\frac{k}{2},\frac{\alpha}{2}}(\mathbb{R}^n) & k\textrm{ even}, \\
C^{\frac{k-1}{2},\frac{1+\alpha}{2}}(\mathbb{R}^n) & k\textrm{ odd}.
  \end{cases} \] We also investigate sufficient conditions for such regularity
preserving decompositions to exist when $k\geq 4$, and we construct examples of
functions which cannot be decomposed into finite sums of half-regular squares.
  The aforementioned result of Fefferman & Phong, as well as its subsequent
refinements by Tataru, Sawyer & Korobenko, and many other authors, have been
repeatedly used to investigate the properties of certain differential
operators. We discuss similar applications of our generalized decomposition
result to several problems in partial differential equations. In addition, we
develop techniques for constructing non-negative polynomials which are not sums
of squares of polynomials, and we prove related results which could not be
found in a review of the literature.
