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.
Let 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
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
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.
- where .
- , where Mj* is the largest log convex sequence bounded above by Mj.
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.
- 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, JSTOR 2315100, 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