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] &sub; 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 &isin; C&infin;([a,b]) which satisfy


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

for all x &isin; [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 &isin; CM([a,b]) and
 * $$\frac{d^k f}{dx^k}(x) = 0$$

for some point x &isin; [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


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

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
 * $$M_rM_s\leq M_{r+s}$$ 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 after  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 $$L_j= \inf_{k\ge j}(k\cdot M_k^{1/k})$$.
 * $$\sum_j \frac{1}{j}(M_j^*)^{-1/j} = \infty$$, 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: 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.

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 $$\sup_{j\geq 1}(M_j)^{1/j}<\infty$$
 * 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 $$g(x_1,x_2,\ldots,x_n)=x_1+x_2^2$$.