Grant: Discovery Grants Program - Individual

This research centers around the theory of complex approximation. This means that we are interested to know if it is possible, given a certain class of functions, to approximate this class by the ""better-behaved"" class of analytic functions. Better-behaved may mean easier to compute, such as polynomials, or simply defined everywhere so that they act as a substitute for an extension. The point of view we take is qualitative rather than quantitative, in the sense that we are more interested to determine if it is possible to do the approximation than to find an efficient algorithm to compute this approximation. One of the practical consequences of qualitative approximation theory is of course that one should not waste time and energy looking for an approximation that does not exist. The theoritical consequences are deep and important, providing a set of fundamental techniques which are used to simplify many arguments in mathematics. For example, to prove that a property holds for a class of functions that we know can be approximated, one may first prove that it holds for the ""better-behaved"" class of approximants and then prove that, as a consequence, it must hold for the functions being approximated. I am especially interested in approximating a function by solutions of a system of differential equations. In particular, holomorphic functions, harmonic functions and other functions which are solutions of an elliptic partial differential equation have been considered and generalizations (with applications) of classical theorems have been obtained.