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 $[a,b]\subset \mathbb {R}$, 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 with $M_{0}=1$. Then we define the class of functions $C^{M}([a,b])$ to be those $f\in C^{\infty }([a,b])$ which satisfy

$\left|{\frac {d^{k}f}{dx^{k}}}(x)\right|\leq C^{k+1}M_{k}$

for all $x\in [a,b]$, some constant C, 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 $f\in C^{M}([a,b])$ and

${\frac {d^{k}f}{dx^{k}}}(x)=0$

for some point $x\in [a,b]$ 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 (1923) 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 $L_{j}=\inf _{k\geq j}M_{k}^{1/k}$
• $\sum _{j}(M_{j}^{*})^{-1/j}=\infty$, where M_j^* is the largest log convex sequence bounded above by M_j.
• $\sum _{j}M_{j-1}^{*}/M_{j}^{*}=\infty$

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

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, 75: 26–31, ISSN 0002-9890, MR0225957
• 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. {{cite book}}: Check |isbn= value: length (help)
• Leont'ev, A.F. (2001) [1994], "Quasi-analytic class", Encyclopedia of Mathematics, EMS Press
• Solomentsev, E.D. (2001) [1994], "Carleman theorem", Encyclopedia of Mathematics, EMS Press