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Quasi analytic function

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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.

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Definitions

Let M = { M k } k = 0 ∞ be a sequence of positive real numbers. Then we define the class of functions CM([a,b]) to be those f ∈ C∞([a,b]) which satisfy

| d k f d x k ( x ) | ≤ A k + 1 M k

for all x ∈ [a,b], some constant A, and all non-negative integers k. If Mk = k! 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

d k f d x k ( x ) = 0

for some point x ∈ [a,b] and all k, f is identically equal to zero.

A function f is called a quasi-analytic function if f is in some quasi-analytic class.

The Denjoy–Carleman theorem

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:

  • CM([a,b]) is quasi-analytic.
  • ∑ 1 / L j = ∞ where L j = inf k ≥ j M k 1 / k .
  • ∑ j ( M j ∗ ) − 1 / j = ∞ , where Mj* is the largest log convex sequence bounded above by Mj.
  • ∑ j M j − 1 ∗ / M j ∗ = ∞ .

    • 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

    n ! , n ! ( ln ⁡ n ) n , n ! ( ln ⁡ n ) n ( ln ⁡ ln ⁡ n ) n , n ! ( ln ⁡ n ) n ( ln ⁡ ln ⁡ n ) n ( ln ⁡ ln ⁡ ln ⁡ n ) n , … ,

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

    References

    Quasi-analytic function Wikipedia
    (Text) CC BY-SA


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