# Quasi-analytic function

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.

## Definitions

Let be a sequence of positive real numbers. Then we define the class of functions *C*^{M}([*a*,*b*]) to be those *f* ∈ *C*^{∞}([*a*,*b*]) which satisfy

for all *x* ∈ [*a*,*b*], some constant *A*, 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* ∈ *C*^{M}([*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 *C*^{M}([*a*,*b*]) is a quasi-analytic class. It states that the following conditions are equivalent:

*C*^{M}([*a*,*b*]) is quasi-analytic.- where .
- , where
*M*_{j}^{*}is the largest log convex sequence bounded above by*M*_{j}.

The proof that the last two conditions are equivalent to the second uses Carleman's inequality.

Example: Template:Harvtxt pointed out that if *M*_{n} is given by one of the sequences

then the corresponding class is quasi-analytic. The first sequence gives analytic functions.

## References

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