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 ⊂ R, and at some point f and all of its derivatives are zero, then f is identically zero on all of . 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 is defined to be those f ∈ C∞ which satisfy for all x ∈ , some constant A, and all non-negative integersk. If Mk = 1 this is exactly the class of real analytic functions on . The class CM is said to be quasi-analytic if whenever f ∈ CM and for some point x ∈ 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.
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 derivativesequal 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 When is logarithmically convex, then is increasing and The quasi-analytic class with respect to a logarithmically convex sequence satisfies:
is closed under composition. Specifically, if and, then.
The Denjoy–Carleman theorem
The Denjoy–Carleman theorem, proved by after gave some partial results, gives criteria on the sequence M under which CM is a quasi-analytic class. It states that the following conditions are equivalent:
CM is quasi-analytic.
where.
, 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
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.
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 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 doesn't satisfy the Weierstrass division property with respect to.
