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

be a sequence of positive real numbers. Then the Denjoy–Carleman class of functions CM([a,b]) is defined to be those fC([a,b]) which satisfy

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 fCM([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

, and

and

such that

.

, although other notations abound.

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

is non-decreasing.

is said to be logarithmically convex, if

1. is increasing.

is increasing and

1. .

satisfies:

• is a ring. In particular it is closed under multiplication.

• .

Denjoy–Carleman theorem

The Denjoy–Carleman theorem, proved by after 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.

• .

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

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

Example: pointed out that if Mn is given by one of the sequences

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

Additional properties

the following properties of the corresponding class of functions hold:

• contains the analytic functions, and it is equal to it if and only if

• .

• .

• For any infinitely differentiable function.

Weierstrass division

is said to be variables is said to satisfy the such that

1. .

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.

.

References