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 CM([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 Mk = 1 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, 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 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.

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