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 $[a,b]\subset \mathbb {R}$ , 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 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^{k}f}{dx^{k}}}(x)\right|\leq C^{k+1}M_{k}

for all $x\in [a,b]$ , 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.

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 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
$\sum 1/L_{j}=\infty$ where $L_{j}=\inf _{k\geq j}M_{k}^{1/k}$
$\sum _{j}(M_{j}^{*})^{-1/j}=\infty$ , where M_j^* is the largest log convex sequence bounded above by M_j.
$\sum _{j}M_{j-1}^{*}/M_{j}^{*}=\infty$