  We develop a convolution theory for quasianalytic ultradistributions of
Gelfand-Shilov type. We also construct a special class of ultrapolynomials, and
use it as a base for the parametrix method in the study of new topological and
structural properties of several quasianalytic spaces of functions and
ultradistributions. In particular, our results apply to Fourier hyperfunctions
and Fourier ultra-hyperfunctions.
