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

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

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

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

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 Carleman (1926) after Denjoy (1921) 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.
• $\sum 1/L_j = \infty$ where $L_j= \inf_{k\ge j}M_k^{1/k}$.
• $\sum_j(M_j^*)^{-1/j} = \infty$, where Mj* is the largest log convex sequence bounded above by Mj.
• $\sum_jM_{j-1}^*/M_j^* = \infty.$

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

$n^n,\,(n\log n)^n,\,(n\log n\log \log n)^n,\,(n\log n\log \log n\log \log \log n)^n\dots$

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