A new Approach to Temperate Generalized Colombeau Functions

Antoine Delcroix

A new approach to the algebra $\Cal G_\tau$ of temperate nonlinear generalized functions is proposed, in which $\Cal G_\tau$ is based on the space $\Cal O_M$ endowed with is natural topology in contrary to previous constructions. Thus, this construction fits perfectly in the general scheme of construction of Colombeau type algebras and reveals better properties of $\Cal G_\tau$. This is illustrated by the natural introduction of a regularity theory in $\Cal G_\tau$, of the Fourier transform, with the definition of $\Cal G_{\Cal O'_C}$, the space of rapidly generalized distributions which is the Fourier image of $\Cal G_\tau$.