Regularity Preserving Sum of Squares Decompositions

Abstract

It is well-known that every non-negative function in C^{3,1}(R^n) can be written as a finite sum of squares of functions in C^{1,1}(R^n). In this thesis, we generalize this decomposition result to show that if f is a non-negative function in C^{k,a}(R^n), where k<4 and 0<a<1, then f can be written as a finite sum of squares of functions that are each `half' as regular as f. By this we mean that the decomposition functions belong to the Holder space of functions which have at least half as many derivatives as f, and their highest order derivatives are all Holder continuous with exponent a/2.

We also investigate sufficient conditions for these regularity preserving decompositions to exist when k>3, and we construct examples of functions which cannot be decomposed into finite sums of half-regular squares.

Existing decomposition results have been 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.