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.
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.
is called quasi-analytic on the open set
The Denjoy-Carleman class of functions of
variables with respect to the sequence
, 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.
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: 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: 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 then the corresponding class is quasi-analytic.
The first sequence gives analytic functions.
For a logarithmically convex sequence
the following properties of the corresponding class of functions hold: A function
is said to be regular of order
regular of order
of real or complex functions of
variables is said to satisfy the Weierstrass division with respect to
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.
is logarithmically convex and
is not equal to the class of analytic function, then
doesn't satisfy the Weierstrass division property with respect to