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| In mathematics, a '''quasi-analytic''' class of '''functions''' is a generalization of the class of real [[analytic function]]s 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.
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| ==Definitions==
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| Let <math>M=\{M_k\}_{k=0}^\infty</math> be a sequence of positive real numbers. Then we define the class of functions ''C''<sup>''M''</sup>([''a'',''b'']) to be those ''f'' ∈ ''C''<sup>∞</sup>([''a'',''b'']) which satisfy
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| :<math>\left |\frac{d^kf}{dx^k}(x) \right | \leq A^{k+1} M_k </math> | |
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| for all ''x'' ∈ [''a'',''b''], some constant ''A'', and all non-negative integers ''k''. If ''M''<sub>''k''</sub> = ''k''! this is exactly the class of real [[analytic function]]s on [''a'',''b'']. The class ''C''<sup>''M''</sup>([''a'',''b'']) is said to be ''quasi-analytic'' if whenever ''f'' ∈ ''C''<sup>''M''</sup>([''a'',''b'']) and
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| :<math>\frac{d^k f}{dx^k}(x) = 0</math>
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| for some point ''x'' ∈ [''a'',''b''] and all ''k'', ''f'' is identically equal to zero.
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| A function ''f'' is called a ''quasi-analytic function'' if ''f'' is in some quasi-analytic class.
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| ==The Denjoy–Carleman theorem==
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| The Denjoy–Carleman theorem, proved by {{harvtxt|Carleman|1926}} after {{harvtxt|Denjoy|1921}} gave some partial results, gives criteria on the sequence ''M'' under which ''C''<sup>''M''</sup>([''a'',''b'']) is a quasi-analytic class. It states that the following conditions are equivalent:
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| *''C''<sup>''M''</sup>([''a'',''b'']) is quasi-analytic.
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| *<math>\sum 1/L_j = \infty</math> where <math>L_j= \inf_{k\ge j}M_k^{1/k}</math>.
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| *<math>\sum_j(M_j^*)^{-1/j} = \infty</math>, where ''M''<sub>''j''</sub><sup>*</sup> is the largest log convex sequence bounded above by ''M''<sub>''j''</sub>.
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| *<math>\sum_jM_{j-1}^*/M_j^* = \infty.</math>
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| The proof that the last two conditions are equivalent to the second uses [[Carleman's inequality]].
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| Example: {{harvtxt|Denjoy|1921}} pointed out that if ''M''<sub>''n''</sub> is given by one of the sequences
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| :<math>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</math>
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| then the corresponding class is quasi-analytic. The first sequence gives analytic functions.
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| ==References==
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| *{{citation|authorlink=Torsten Carleman|first= T. |last=Carleman|title=Les fonctions quasi-analytiques|publisher= Gauthier-Villars |year=1926}}
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| *{{Citation | doi=10.2307/2315100 | authorlink=Paul Cohen (mathematician) | last1=Cohen | first1=Paul J. | title=A simple proof of the Denjoy-Carleman theorem | url=http://www.jstor.org/stable/2315100 | id={{MathSciNet | id = 0225957}} | year=1968 | journal=[[American Mathematical Monthly|The American Mathematical Monthly]] | issn=0002-9890 | volume=75 | pages=26–31 | issue=1 | publisher=Mathematical Association of America}}
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| *{{citation|authorlink=Arnaud Denjoy|first= A. |last=Denjoy|title=Sur les fonctions quasi-analytiques de variable réelle|journal= C.R. Acad. Sci. Paris |volume= 173 |year=1921|pages= 1329–1331}}
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| *{{Citation|first=Lars|last=[[Lars Hörmander|Hörmander]]|title=The Analysis of Linear Partial Differential Operators I|publisher=Springer-Verlag|year=1990|isbn=3-540-00662}}
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| *{{eom|id=Q/q076370|title=Quasi-analytic class|first=A.F.|last= Leont'ev}}
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| *{{eom|id=C/c020430|title=Carleman theorem|first=E.D.|last= Solomentsev}}
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| [[Category:Smooth functions]]
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