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

| \frac{d^{k} f}{d x^{k}} (x) | \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}{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 $f : ℝ^{n} \to ℝ$ and multi-indexes $j = (j_{1}, j_{2}, \dots, j_{n}) \in ℕ^{n}$ , 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 $U \subset ℝ^{n}$ if for every compact $K \subset U$ there is a constant $A$ such that

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

for all multi-indexes $j \in ℕ^{n}$ 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 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.
$\sum 1 / L_{j} = \infty$ 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 $g : ℝ^{n} \to ℝ$ 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 $h_{1}, h_{2}, \dots, h_{d - 1} \in A_{n - 1}$ such that

f = g q + 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 $C^{M}$ 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, 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