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 , and at some point f and all of its derivatives are zero, then f is identically zero on all of . Quasi-analytic classes are broader classes of functions for which this statement still holds true.
Let be a sequence of positive real numbers with . Then we define the class of functions to be those which satisfy
for all , some constant C, and all non-negative integers k. If this is exactly the class of real analytic functions on . The class is said to be quasi-analytic if whenever and
for some point 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, proved by Carleman (1926) after Denjoy (1921) gave some partial results, gives criteria on the sequence M under which is a quasi-analytic class. It states that the following conditions are equivalent:
The proof that the last two conditions are equivalent to the second uses Carleman's inequality.
Example: Denjoy (1921) 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.