# 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 the Denjoy-Carleman class of functions CM([a,b]) is defined to be those f ∈ C([a,b]) which satisfy

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

for all x ∈ [a,b], some constant A, and all non-negative integers k. If Mk = 1 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, then f is identically equal to zero.

A function f is called a quasi-analytic function if f is in some quasi-analytic class.

### Quasi-analytic functions of several variables

For a function $f:\mathbb {R} ^{n}\to \mathbb {R}$  and multi-indexes $j=(j_{1},j_{2},\ldots ,j_{n})\in \mathbb {N} ^{n}$ , denote $|j|=j_{1}+j_{2}+\ldots +j_{n}$ , and

$D^{j}={\frac {\partial ^{j}}{\partial x_{1}^{j_{1}}\partial x_{2}^{j_{2}}\ldots \partial x_{n}^{j_{n}}}}$
$j!=j_{1}!j_{2}!\ldots j_{n}!$

and

$x^{j}=x_{1}^{j_{1}}x_{2}^{j_{2}}\ldots x_{n}^{j_{n}}.$

Then $f$  is called quasi-analytic on the open set $U\subset \mathbb {R} ^{n}$  if for every compact $K\subset U$  there is a constant $A$  such that

$\left|D^{j}f(x)\right|\leq A^{|j|+1}j!M_{|j|}$

for all multi-indexes $j\in \mathbb {N} ^{n}$  and all points $x\in K$ .

The Denjoy-Carleman class of functions of $n$  variables with respect to the sequence $M$  on the set $U$  can be denoted $C_{n}^{M}(U)$ , although other notations abound.

The Denjoy-Carleman class $C_{n}^{M}(U)$  is said to be quasi-analytic when the only function in it having all its partial derivatives equal to zero at a point is the function identically equal to zero.

A function of several variables is said to be quasi-analytic when it belongs to a quasi-analytic Denjoy-Carleman class.

### Quasi-analytic classes with respect to logarithmically convex sequences

In the definitions above it is possible to assume that $M_{1}=1$  and that the sequence $M_{k}$  is non-decreasing.

The sequence $M_{k}$  is said to be logarithmically convex, if

$M_{k+1}/M_{k}$  is increasing.

When $M_{k}$  is logarithmically convex, then $(M_{k})^{1/k}$  is increasing and

$M_{r}M_{s}\leq M_{r+s}$  for all $(r,s)\in \mathbb {N} ^{2}$ .

The quasi-analytic class $C_{n}^{M}$  with respect to a logarithmically convex sequence $M$  satisfies:

• $C_{n}^{M}$  is a ring. In particular it is closed under multiplication.
• $C_{n}^{M}$  is closed under composition. Specifically, if $f=(f_{1},f_{2},\ldots f_{p})\in (C_{n}^{M})^{p}$  and $g\in C_{p}^{M}$ , then $g\circ f\in C_{n}^{M}$ .

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

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

$1,\,{(\ln n)}^{n},\,{(\ln n)}^{n}\,{(\ln \ln n)}^{n},\,{(\ln n)}^{n}\,{(\ln \ln n)}^{n}\,{(\ln \ln \ln n)}^{n},\dots ,$

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

For a logarithmically convex sequence $M$  the following properties of the corresponding class of functions hold:

• $C^{M}$  contains the analytic functions, and it is equal to it if and only if $\sup _{j\geq 1}(M_{j})^{1/j}<\infty$
• If $N$  is another logarithmically convex sequence, with $M_{j}\leq C^{j}N_{j}$  for some constant $C$ , then $C^{M}\subset C^{N}$ .
• $C^{M}$  is stable under differentiation if and only if $\sup _{j\geq 1}(M_{j+1}/M_{j})^{1/j}<\infty$ .
• For any infinitely differentiable function $f$  there are quasi-analytic rings $C^{M}$  and $C^{N}$  and elements $g\in C^{M}$ , and $h\in C^{N}$ , such that $f=g+h$ .

### Weierstrass division

A function $g:\mathbb {R} ^{n}\to \mathbb {R}$  is said to be regular of order $d$  with respect to $x_{n}$  if $g(0,x_{n})=h(x_{n})x_{n}^{d}$  and $h(0)\neq 0$ . Given $g$  regular of order $d$  with respect to $x_{n}$ , a ring $A_{n}$  of real or complex functions of $n$  variables is said to satisfy the Weierstrass division with respect to $g$  if for every $f\in A_{n}$  there is $q\in A$ , and $h_{1},h_{2},\ldots ,h_{d-1}\in A_{n-1}$  such that

$f=gq+h$  with $h(x',x_{n})=\sum _{j=0}^{d-1}h_{j}(x')x_{n}^{j}$ .

While the ring of analytic functions and the ring of formal power series both satisfy the Weierstrass division property, the same is not true for other quasi-analytic classes.

If $M$  is logarithmically convex and $C^{M}$  is not equal to the class of analytic function, then $C^{M}$  doesn't satisfy the Weierstrass division property with respect to $g(x_{1},x_{2},\ldots ,x_{n})=x_{1}+x_{2}^{2}$ .