Quasi-analytic functions of several variables

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

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

${\displaystyle j!=j_{1}!j_{2}!\ldots j_{n}!}$

and

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

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

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

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

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

The Denjoy-Carleman class ${\displaystyle C_{n}^{M}(U)}$ 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 ${\displaystyle M_{1}=1}$ and that the sequence ${\displaystyle M_{k}}$ is non-decreasing.

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

${\displaystyle M_{k+1}/M_{k}}$ is increasing.

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

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

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

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

Weierstrass division

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

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

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