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.

Definitions[edit]

Let M=\{M_k\}_{k=0}^\infty 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

\left |\frac{d^kf}{dx^k}(x) \right | \leq 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

\frac{d^k f}{dx^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[edit]

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.
  • \sum 1/L_j = \infty where L_j= \inf_{k\ge j}M_k^{1/k}.
  • \sum_j(M_j^*)^{-1/j} = \infty, where Mj* is the largest log convex sequence bounded above by Mj.
  • \sum_jM_{j-1}^*/M_j^* = \infty.

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,\,(n\log n)^n,\,(n\log n\log \log n)^n,\,(n\log n\log \log n\log \log \log n)^n\dots

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

References[edit]