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 C^M([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 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

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

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

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

Example: Denjoy (1921) pointed out that if M_n 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 $g(x_{1},x_{2},\ldots ,x_{n})=x_{1}+x_{2}^{2}$.

References

• Carleman, T. (1926), Les fonctions quasi-analytiques, Gauthier-Villars
• Cohen, Paul J. (1968), "A simple proof of the Denjoy-Carleman theorem", The American Mathematical Monthly, 75 (1), Mathematical Association of America: 26–31, doi:10.2307/2315100, ISSN 0002-9890, JSTOR 2315100, MR 0225957
• Denjoy, A. (1921), "Sur les fonctions quasi-analytiques de variable réelle", C. R. Acad. Sci. Paris, 173: 1329–1331
• Hörmander, Lars (1990), The Analysis of Linear Partial Differential Operators I, Springer-Verlag, ISBN 3-540-00662-1
• Leont'ev, A.F. (2001) [1994], "Quasi-analytic class", Encyclopedia of Mathematics, EMS Press
• Solomentsev, E.D. (2001) [1994], "Carleman theorem", Encyclopedia of Mathematics, EMS Press