Quasi-analytic function

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 $M=\{M_k\}_{k=0}^\infty$ be a sequence of positive real numbers. Then we define the class of functions C^M([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 M_k = k! this is exactly the class of real analytic functions on [a,b]. The class C^M([a,b]) is said to be quasi-analytic if whenever f ∈ C^M([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

The Denjoy–Carleman theorem, proved by Carleman (1926) after Denjoy (1921) gave some partial results, gives criteria on the sequence M under which C^M([a,b]) is a quasi-analytic class. It states that the following conditions are equivalent:

• C^M([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 M_j^* is the largest log convex sequence bounded above by M_j.
• $\sum_jM_{j-1}^*/M_j^* = \infty.$

The proof that the last two conditions are equivalent to the second uses Carleman's inequality.

Example: Denjoy (1921) pointed out that if M_n is given by one of the sequences

$n!,\, n!\,{(\ln n)}^n,\, n!\,{(\ln n)}^n\,{(\ln \ln n)}^n,\, n!\,{(\ln n)}^n\,{(\ln \ln n)}^n\,{(\ln \ln \ln n)}^n, \dots,$

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

References

• Carleman, T. (1926), Les fonctions quasi-analytiques, Gauthier-Villars
• Cohen, Paul J. (1968), "A simple proof of the Denjoy-Carleman theorem", The American Mathematical Monthly (Mathematical Association of America) 75 (1): 26–31, doi:10.2307/2315100, ISSN 0002-9890, MR 0225957
• Denjoy, A. (1921), "Sur les fonctions quasi-analytiques de variable réelle", C.R. Acad. Sci. Paris 173: 1329–1331
• Hörmander, Lars (1990), The Analysis of Linear Partial Differential Operators I, Springer-Verlag, ISBN 3-540-00662-1
• Leont'ev, A.F. (2001), "Quasi-analytic class", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
• Solomentsev, E.D. (2001), "Carleman theorem", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4