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 ${\displaystyle M=\{M_{k}\}_{k=0}^{\infty }}$ 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

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

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

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

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 ${\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} }$ 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 ${\displaystyle g\circ f\in C_{n}^{M}}$.

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

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

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

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

Additional properties

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

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

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 ${\displaystyle g(x_{1},x_{2},\ldots ,x_{n})=x_{1}+x_{2}^{2}}$.

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, Mathematical Association of America, 75 (1): 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