In mathematics, a quasi-analytic class of functions is a generalization of the class of real analytic functions based upon the following fact. If f is an analytic function on an interval [a,b] ⊂ R, and at some point f and all of its derivatives 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.
for all x ∈ [a,b], some constant A, and all non-negative integers k. If Mk = k! this is exactly the class of real analytic functions on [a,b]. The class CM([a,b]) is said to be quasi-analytic if whenever f ∈ CM([a,b]) and
for some point x ∈ [a,b] and all k, f is identically equal to zero.
A function f is called a quasi-analytic function if f is in some quasi-analytic class.
The Denjoy–Carleman theorem
The Denjoy–Carleman theorem, proved by Template:Harvtxt after Template:Harvtxt gave some partial results, gives criteria on the sequence M under which CM([a,b]) is a quasi-analytic class. It states that the following conditions are equivalent:
- CM([a,b]) is quasi-analytic.
- where .
- , where Mj* is the largest log convex sequence bounded above by Mj.
The proof that the last two conditions are equivalent to the second uses Carleman's inequality.
Example: Template:Harvtxt pointed out that if Mn is given by one of the sequences
then the corresponding class is quasi-analytic. The first sequence gives analytic functions.