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 ${\displaystyle M=\{M_{k}\}_{k=0}^{\infty }}$ 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

${\displaystyle \left|{\frac {d^{k}f}{dx^{k}}}(x)\right|\leq A^{k+1}M_{k}}$

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

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

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

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

${\displaystyle n!,\,n!\,{(\ln n)}^{n},\,n!\,{(\ln n)}^{n}\,{(\ln \ln n)}^{n},\,n!\,{(\ln n)}^{n}\,{(\ln \ln n)}^{n}\,{(\ln \ln \ln n)}^{n},\dots ,}$

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