Denjoy-Carleman Theorem

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''nbsp;⊂ 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 $M=\_^\infty$ be a sequence of positive real numbers. Then the Denjoy-Carleman class of functions ''C''^''M''( 'a'',''b'' is defined to be those ''f'' ∈ ''C''^∞( 'a'',''b'' which satisfy

:$\left , \frac(x) \right , \leq A^ k! M_k$

for all ''x'' ∈  'a'',''b'' some constant ''A'', and all non-negative integers ''k''. If ''M''_''k'' = 1 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
:$\frac(x) = 0$
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 $f:\mathbb^n\to\mathbb$ and multi-indexes $j=(j_1,j_2,\ldots,j_n)\in\mathbb^n$, denote $, j, =j_1+j_2+\ldots+j_n$, and
:$D^j=\frac$
:$j!=j_1!j_2!\ldots j_n!$
and
:$x^j=x_1^x_2^\ldots x_n^.$

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

:$\left, D^jf(x)\\leq A^j!M_$

for all multi-indexes $j\in\mathbb^n$ 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
:$M_/M_k$ is increasing.

When $M_k$ is logarithmically convex, then $(M_k)^$ is increasing and
:$M_rM_s\leq M_$ for all $(r,s)\in\mathbb^2$.

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 after 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.
*$\sum 1/L_j = \infty$ where $L_j= \inf_(k\cdot M_k^)$.
*$\sum_j \frac(M_j^*)^ = \infty$, where ''M''_''j''^* is the largest log convex sequence bounded above by ''M''_''j''.
*$\sum_j\frac = \infty.$

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

Example: pointed out that if ''M''_''n'' is given by one of the sequences
:$1,\, ^n,\, ^n\,^n,\, ^n\,^n\,^n, \dots,$
then the corresponding class is quasi-analytic. The first sequence gives analytic functions.

Additional properties

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 $\sup_(M_j)^<\infty$
* 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 $\sup_(M_/M_j)^<\infty$.
* 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 $g:\mathbb^n\to\mathbb$ 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 $h_1,h_2,\ldots,h_\in A_$ such that

:$f=gq+h$ with $h(x',x_n)=\sum_^h_(x')x_n^j$.

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 $g(x_1,x_2,\ldots,x_n)=x_1+x_2^2$.

References

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*{{eom, id=C/c020430, title=Carleman theorem, first=E.D., last= Solomentsev

Smooth functions