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 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

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

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 and multi-indexes , denote , and

and

Then is called quasi-analytic on the open set if for every compact there is a constant such that

for all multi-indexes and all points .

The Denjoy–Carleman class of functions of variables with respect to the sequence on the set can be denoted , although other notations abound.

The Denjoy–Carleman class 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 and that the sequence is non-decreasing.

The sequence is said to be logarithmically convex, if

is increasing.

When is logarithmically convex, then is increasing and

for all .

The quasi-analytic class with respect to a logarithmically convex sequence satisfies:

• is a ring. In particular it is closed under multiplication.
• is closed under composition. Specifically, if and , then .

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.
• where .
• , where M_j^* is the largest log convex sequence bounded above by M_j.

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

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

Additional properties

For a logarithmically convex sequence the following properties of the corresponding class of functions hold:

• contains the analytic functions, and it is equal to it if and only if
• If is another logarithmically convex sequence, with for some constant , then .
• is stable under differentiation if and only if .
• For any infinitely differentiable function there are quasi-analytic rings and and elements , and , such that .

Weierstrass division

A function is said to be regular of order with respect to if and . Given regular of order with respect to , a ring of real or complex functions of variables is said to satisfy the Weierstrass division with respect to if for every there is , and such that

with .

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 is logarithmically convex and is not equal to the class of analytic function, then does not satisfy the Weierstrass division property with respect to .

References