User:Holmansf/Quasi-analytic classes

A quasi-analytic class of functions is a generalization of the class of analytic functions based upon the following fact. If f is an analytic function on an interval $$[a,b] \subset \mathbb{R} $$, and at some point f and all of its deriviates are zero, then f is identically zero on all of $$[a,b]$$. Quasi-analytic classes are broader classes of functions for which this statement still holds true.

Definitions
Let $$M = \{ M_k \}_{k=0}^\infty$$ be a sequence of positive real numbers with $$M_0 = 1$$. Then we define the class of functions $$C^M([a,b])$$ to be those $$f \in C^\infty([a,b])$$ which satisfy


 * $$\left |\frac{d^kf}{dx^k}(x) \right | \leq C^{k+1} M_k $$

for some constant C and all non-negative integers k. If $$M_k = k!$$ this is exactly the class of real-analytic functions on $$[a,b]$$. The class $$C^M([a,b])$$ is said to be quasi-analytic if whenever $$f \in C^M([a,b])$$ and


 * $$\frac{d^k f}{dx^k}(x) = 0$$

for some point $$x \in [a,b]$$ and all k, f is identically equal to zero. The Denjoy-Carleman theorem gives criteria on the sequence M under which $$C^M([a,b])$$ is a quasi-analytic class.

A function f is called a quasi-analytic function if f is in some quasi-analytic class.