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 $M=\{M_k{\}}_{k=0}^{\mathrm{\infty }}$ 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

$|\frac{d^{k}f}{d{x}^k}(x)|\le A^{k+1}k!M_k$

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

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

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 $j=(j_1,j_2,\dots ,j_n)\in {\mathbb{\mathbb{N}}}^n$, denote $|j|=j_1+j_2+\dots +j_n$, and

$D^j=\frac{{\partial }^j}{\partial x_1^{j_1}\partial x_2^{j_2}\dots \partial x_n^{j_n}}$

$j!=j_1!j_2!\dots j_n!$

and

$x^j=x_1^{j_1}{x}_2^{j_2}\dots x_n^{j_n}.$

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

$|D^{j}f(x)|\le A^{|j|+1}j!M_{|j|}$

for all multi-indexes $j\in {\mathbb{\mathbb{N}}}^n$ 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

$M_r{M}_s\le M_{r+s}$ for all $(r,s)\in {\mathbb{\mathbb{N}}}^2$.

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 $f=(f_1,f_2,\dots f_p)\in (C_n^M{)}^p$ and $g\in C_p^M$, then $g\circ f\in C_n^M$.

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.
• $\sum 1/L_j=\mathrm{\infty }$ where $L_j=\underset{k\ge j}{\inf }(k\cdot M_k^{1/k})$.
• $\sum _j\frac{1}{j}(M_j^{*}{)}^{-1/j}=\mathrm{\infty }$, where M_j^* is the largest log convex sequence bounded above by M_j.
• $\sum _j\frac{M_{j-1}^{*}}{(j+1)M_j^{*}}=\mathrm{\infty }.$

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 $\underset{j\ge 1}{\sup }(M_j{)}^{1/j}<\mathrm{\infty }$
• If $N$ is another logarithmically convex sequence, with $M_j\le C^j{N}_j$ for some constant $C$, then $C^M\subset C^N$.
• $C^M$ is stable under differentiation if and only if $\underset{j\ge 1}{\sup }(M_{j+1}/M_j{)}^{1/j}<\mathrm{\infty }$.
• 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 $h(0)\ne 0$. 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 $h_1,h_2,\dots ,h_{d-1}\in A_{n-1}$ such that

$f=gq+h$ with $h(x^{\prime },x_n)=\sum _{j=0}^{d-1}{h}_j(x^{\prime })x_n^j$.

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 $C^M$ does not satisfy the Weierstrass division property with respect to $g(x_1,x_2,\dots ,x_n)=x_1+x_2^2$.

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 
• 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 
• Hazewinkel, Michiel, ed. (2001), "Quasi-analytic class", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=Q/q076370 
• Hazewinkel, Michiel, ed. (2001), "Carleman theorem", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=C/c020430