Quasi-analytic function

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^kf}{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^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 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\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 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: 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}
• 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}.
• 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