Carleman theorem
Carleman's theorem on quasi-analytic classes of functions is a necessary and sufficient condition for quasi-analyticity in the sense of Hadamard, discovered by T. Carleman
(see also ). A class of real-valued infinitely differentiable functions
on an interval
is said to be quasi-analytic in the sense of Hadamard if the equalities
,
at some fixed point
,
, imply that
. The statement of the theorem: The class
is quasi-analytic if and only if
![]() | (1) |
where
![]() |
is a constant, and the sequence
satisfies one of the equivalent conditions:
![]() | (2) |
![]() |
where
![]() |
This is one of the first definitive results in the theory of quasi-analytic classes of functions. Quasi-analytic classes defined by (1), (2) are often called Carleman classes.
Carleman's theorem on conditions of well-definedness of moment problems: If the sequence of positive numbers ,
satisfies the condition
![]() |
then the moment problem
![]() | (3) |
is well-defined. This means that there exists a non-decreasing function ,
, satisfying the equations (3), which is unique up to addition by any function which is constant in a neighbourhood of each point of continuity of it. This theorem was established by T. Carleman (see , ).
Carleman's theorem on uniform approximation by entire functions: If is any continuous function on the real line and
,
, is a positive continuous function decreasing arbitrarily rapidly as
, then there exists an entire function
of the complex variable
such that
![]() |
This theorem, established by T. Carleman , was the starting point in the investigations into approximation by entire functions. In particular, a continuum in the
-plane is said to be a Carleman continuum if for any continuous complex function
on
and an arbitrary rapidly decreasing positive function
(as
) with a positive infimum on any finite interval, there exists an entire function
such that
![]() |
Necessary and sufficient conditions for a closed set to be a Carleman continuum were obtained in a theorem by M.V. Keldysh and M.A. Lavrent'ev (see ). An example of a Carleman continuum is a closed set consisting of rays of the form
![]() |
Carleman's theorem on the approximation of analytic functions by polynomials in the mean over the area of a domain: Let be a finite domain in the complex
-plane,
, bounded by a Jordan curve
, and let
be a regular analytic function in
such that
![]() |
Then there exists for any a polynomial
such that
![]() |
This result was established by T. Carleman [4]. Similar results also hold for approximation with an arbitrary positive continuous weight, in which case the boundary can be of a more general nature. The system of monomials
,
is complete with respect to any such weight. Orthogonalization and normalization of this system gives polynomials
of degree
, which are often called Carleman polynomials.
References
[1] | T. Carleman, "Les fonctions quasi-analytiques" , Gauthier-Villars (1926) |
[2] | T. Carleman, "Sur les équations intégrales singulières à noyau réel et symmétrique" Univ. Årsskrift : 3 , Uppsala (1923) |
[3] | T. Carleman, "Sur un théorème de Weierstrass" Arkiv. Mat. Astron. Fys. , 20 : 4 (1927) pp. 1–5 |
[4] | T. Carleman, "Über die Approximation analytischer Funktionen durch lineare Aggregate von vorgegebenen Potenzen" Arkiv. Mat. Astron. Fys. , 17 : 9 (1922) |
[5] | S. Mandelbrojt, "Séries adhérentes, régularisations des suites, applications" , Gauthier-Villars (1952) |
[6] | S.N. Mergelyan, "Uniform approximation to functions of a complex variable" Translations Amer. Math. Soc. , 3 (1962) pp. 294–391 Uspekhi Mat. Nauk , 7 : 2 (1952) pp. 31–122 |
Comments
The following result is also known as Carleman's theorem. If is a holomorphic function in the region
![]() |
and ,
, are the zeros of
(counted with multiplicity) in
, then
![]() |
![]() |
where
![]() |
See [a2]. Further, [a1] is a good reference for the approximation theorems in the present article.
References
[a1] | D. Gaier, "Vorlesungen über Approximation im Komplexen" , Birkhäuser (1980) |
[a2] | B.Ya. Levin, "Distribution of zeros of entire functions" , Amer. Math. Soc. (1980) (Translated from Russian) |
Carleman theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Carleman_theorem&oldid=16318