Denjoy-Carleman classes of differentiable functions, say in Roumieu's form:
Given a log-convex sequence $M_n$ of positive number denote by $C_M=C_M(\mathbb{R}^n,0)$ the ring of germs of $C^\infty(\mathbb{R}^n)$ functions such that for every compact $K$ there are constants $A$ and $B$ such that $|D^{\alpha}f(x)|\leq AB^{|\alpha|}M_{|\alpha|}$ for every x in $K$ and every multi-index $\alpha$.
In what ways are the $C_M$ important as a classes? Especially in the case in which of $C_M$ is quasianalytic. These sets of smooth functions satisfy a number of properties (perhaps adding extra conditions on the sequence $M$): are local rings, closed under composition, differentiation, implicit function theorem, ... These properties make them important, and the definition is perhaps natural, so that makes them important in a sense. Now, is there some context in which $C_M$ are precisely the solution to some problem? This is what I mean by importance of $C_M$ as a class.
Suppose we allocate all quasianalytic functions (all functions belonging to some quasianalytic $C_M$ for some $M$) in classes in some other way that still satisfy all desirable properties above and some others. If there is some sense in which the $C_M$ is specially relevant as a class it is not to be expected that some new subdivision of the quasianalytic functions is also going to have that property.
[I put tags according to where I imagine there are experts with an answer to this question, instead of just the nature of the objects in the question. Is this a valid way of using tags?]
