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^{k}f}{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 $L_{j}=\inf _{{k\geq j}}M_{k}^{{1/k}}$ .
$\sum _{j}(M_{j}^{*})^{{-1/j}}=\infty$ , 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.

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, 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

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