Definitions
Let be a sequence of positive real numbers. Then the Denjoy-Carleman class of functions ''C''''M''( 'a'',''b'' is defined to be those ''f'' ∈ ''C''∞( 'a'',''b'' which satisfy : for all ''x'' ∈ 'a'',''b'' some constant ''A'', and all non-negative integers ''k''. If ''M''''k'' = 1 this is exactly the class of realQuasi-analytic functions of several variables
For a function and multi-indexes , denote , and : : and : Then is called quasi-analytic on the open set if for every compact there is a constant such that : for all multi-indexes and all points . The Denjoy-Carleman class of functions of variables with respect to the sequence on the set can be denoted , although other notations abound. The Denjoy-Carleman class is said to be quasi-analytic when the only function in it having all its partial derivatives equal to zero at a point is the function identically equal to zero. A function of several variables is said to be quasi-analytic when it belongs to a quasi-analytic Denjoy-Carleman class.Quasi-analytic classes with respect to logarithmically convex sequences
In the definitions above it is possible to assume that and that the sequence is non-decreasing. The sequence is said to be ''logarithmically convex'', if : is increasing. When is logarithmically convex, then is increasing and : for all . The quasi-analytic class with respect to a logarithmically convex sequence satisfies: * is a ring. In particular it is closed under multiplication. * is closed under composition. Specifically, if and , then .The Denjoy–Carleman theorem
The Denjoy–Carleman theorem, proved by after 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. * where . *, where ''M''''j''* is the largest log convex sequence bounded above by ''M''''j''. * The proof that the last two conditions are equivalent to the second usesAdditional properties
For a logarithmically convex sequence the following properties of the corresponding class of functions hold: * contains the analytic functions, and it is equal to it if and only if * If is another logarithmically convex sequence, with for some constant , then . * is stable under differentiation if and only if . * For any infinitely differentiable function there are quasi-analytic rings and and elements , and , such that .Weierstrass division
A function is said to be ''regular of order with respect to '' if and . Given regular of order with respect to , a ring of real or complex functions of variables is said to satisfy the ''Weierstrass division with respect to '' if for every there is , and such that : with . While the ring of analytic functions and the ring of formal power series both satisfy the Weierstrass division property, the same is not true for other quasi-analytic classes. If is logarithmically convex and is not equal to the class of analytic function, then doesn't satisfy the Weierstrass division property with respect to .References
* * * * * *{{eom, id=C/c020430, title=Carleman theorem, first=E.D., last= Solomentsev Smooth functions