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 CM([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 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 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 f is called quasi-analytic on the open set if for every compact K\subset U there is a constant A such that for all multi-indexes 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 When M_k is logarithmically convex, then (M_k)^{1/k} is increasing and The quasi-analytic class C_n^M with respect to a logarithmically convex sequence M satisfies:

The 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: 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

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

Weierstrass division

A function is said to be regular of order d with respect to x_n if 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 such that 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.