# 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.

## DefinitionsEdit

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 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

for some point *x* ∈ [*a*,*b*] and all *k*, then *f* is identically equal to zero.

A function *f* is called a *quasi-analytic function* if *f* is in some quasi-analytic class.

### Quasi-analytic functions of several variablesEdit

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 sequencesEdit

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 theoremEdit

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.- 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 uses Carleman's inequality.

Example: Denjoy (1921) pointed out that if *M*_{n} is given by one of the sequences

then the corresponding class is quasi-analytic. The first sequence gives analytic functions.

## Additional propertiesEdit

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 divisionEdit

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 .

## ReferencesEdit

- 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) [1994], "Quasi-analytic class", in Hazewinkel, Michiel (ed.),
*Encyclopedia of Mathematics*, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4 - Solomentsev, E.D. (2001) [1994], "Carleman theorem", in Hazewinkel, Michiel (ed.),
*Encyclopedia of Mathematics*, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4