Quasi-analytic function

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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 [math]\displaystyle{ M=\{M_k\}_{k=0}^\infty }[/math] be a sequence of positive real numbers. Then the Denjoy-Carleman class of functions CM([a,b]) is defined to be those f ∈ C∞([a,b]) which satisfy

[math]\displaystyle{ \left |\frac{d^kf}{dx^k}(x) \right | \leq A^{k+1} k! M_k }[/math]

for all x ∈ [a,b], some constant A, and all non-negative integers k. If Mk = 1 this is exactly the class of real analytic functions on [a,b].

The class CM([a,b]) is said to be quasi-analytic if whenever f ∈ CM([a,b]) and

[math]\displaystyle{ \frac{d^k f}{dx^k}(x) = 0 }[/math]

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 [math]\displaystyle{ f:\mathbb{R}^n\to\mathbb{R} }[/math] and multi-indexes [math]\displaystyle{ j=(j_1,j_2,\ldots,j_n)\in\mathbb{N}^n }[/math], denote [math]\displaystyle{ |j|=j_1+j_2+\ldots+j_n }[/math], and

[math]\displaystyle{ D^j=\frac{\partial^j}{\partial x_1^{j_1}\partial x_2^{j_2}\ldots\partial x_n^{j_n}} }[/math]
[math]\displaystyle{ j!=j_1!j_2!\ldots j_n! }[/math]

and

[math]\displaystyle{ x^j=x_1^{j_1}x_2^{j_2}\ldots x_n^{j_n}. }[/math]

Then [math]\displaystyle{ f }[/math] is called quasi-analytic on the open set [math]\displaystyle{ U\subset\mathbb{R}^n }[/math] if for every compact [math]\displaystyle{ K\subset U }[/math] there is a constant [math]\displaystyle{ A }[/math] such that

[math]\displaystyle{ \left|D^jf(x)\right|\leq A^{|j|+1}j!M_{|j|} }[/math]

for all multi-indexes [math]\displaystyle{ j\in\mathbb{N}^n }[/math] and all points [math]\displaystyle{ x\in K }[/math].

The Denjoy-Carleman class of functions of [math]\displaystyle{ n }[/math] variables with respect to the sequence [math]\displaystyle{ M }[/math] on the set [math]\displaystyle{ U }[/math] can be denoted [math]\displaystyle{ C_n^M(U) }[/math], although other notations abound.

The Denjoy-Carleman class [math]\displaystyle{ C_n^M(U) }[/math] 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 [math]\displaystyle{ M_1=1 }[/math] and that the sequence [math]\displaystyle{ M_k }[/math] is non-decreasing.

The sequence [math]\displaystyle{ M_k }[/math] is said to be logarithmically convex, if

[math]\displaystyle{ M_{k+1}/M_k }[/math] is increasing.

When [math]\displaystyle{ M_k }[/math] is logarithmically convex, then [math]\displaystyle{ (M_k)^{1/k} }[/math] is increasing and

[math]\displaystyle{ M_rM_s\leq M_{r+s} }[/math] for all [math]\displaystyle{ (r,s)\in\mathbb{N}^2 }[/math].

The quasi-analytic class [math]\displaystyle{ C_n^M }[/math] with respect to a logarithmically convex sequence [math]\displaystyle{ M }[/math] satisfies:

  • [math]\displaystyle{ C_n^M }[/math] is a ring. In particular it is closed under multiplication.
  • [math]\displaystyle{ C_n^M }[/math] is closed under composition. Specifically, if [math]\displaystyle{ f=(f_1,f_2,\ldots f_p)\in (C_n^M)^p }[/math] and [math]\displaystyle{ g\in C_p^M }[/math], then [math]\displaystyle{ g\circ f\in C_n^M }[/math].

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 CM([a,b]) is a quasi-analytic class. It states that the following conditions are equivalent:

  • CM([a,b]) is quasi-analytic.
  • [math]\displaystyle{ \sum 1/L_j = \infty }[/math] where [math]\displaystyle{ L_j= \inf_{k\ge j}(k\cdot M_k^{1/k}) }[/math].
  • [math]\displaystyle{ \sum_j \frac{1}{j}(M_j^*)^{-1/j} = \infty }[/math], where Mj* is the largest log convex sequence bounded above by Mj.
  • [math]\displaystyle{ \sum_j\frac{M_{j-1}^*}{(j+1)M_j^*} = \infty. }[/math]

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

Example: (Denjoy 1921) pointed out that if Mn is given by one of the sequences

[math]\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, }[/math]

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

Additional properties

For a logarithmically convex sequence [math]\displaystyle{ M }[/math] the following properties of the corresponding class of functions hold:

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

Weierstrass division

A function [math]\displaystyle{ g:\mathbb{R}^n\to\mathbb{R} }[/math] is said to be regular of order [math]\displaystyle{ d }[/math] with respect to [math]\displaystyle{ x_n }[/math] if [math]\displaystyle{ g(0,x_n)=h(x_n)x_n^d }[/math] and [math]\displaystyle{ h(0)\neq 0 }[/math]. Given [math]\displaystyle{ g }[/math] regular of order [math]\displaystyle{ d }[/math] with respect to [math]\displaystyle{ x_n }[/math], a ring [math]\displaystyle{ A_n }[/math] of real or complex functions of [math]\displaystyle{ n }[/math] variables is said to satisfy the Weierstrass division with respect to [math]\displaystyle{ g }[/math] if for every [math]\displaystyle{ f\in A_n }[/math] there is [math]\displaystyle{ q\in A }[/math], and [math]\displaystyle{ h_1,h_2,\ldots,h_{d-1}\in A_{n-1} }[/math] such that

[math]\displaystyle{ f=gq+h }[/math] with [math]\displaystyle{ h(x',x_n)=\sum_{j=0}^{d-1}h_{j}(x')x_n^j }[/math].

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 [math]\displaystyle{ M }[/math] is logarithmically convex and [math]\displaystyle{ C^M }[/math] is not equal to the class of analytic function, then [math]\displaystyle{ C^M }[/math] doesn't satisfy the Weierstrass division property with respect to [math]\displaystyle{ g(x_1,x_2,\ldots,x_n)=x_1+x_2^2 }[/math].

References



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