On classification and representation of infinitely differentiable functions

In this paper a classification of infinitely differentiable functions is developed, based on derivatives introduced earlier by the author. In terms of the classes $C_{y,x}(0,u)$ and $C_{y,x}(0,\infty)$ considered in this paper, not only is an interrelation between the different classes given, but also the problem of representation of the functions of these classes, in the case of quasi-analyticity (in the sense of Denjoy and Carleman) as well as in the case of non-quasi-analyticity of the class, is completely solved. Some applications of these results are given.