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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  $  K $
of real-valued infinitely differentiable functions  $  f $
on an interval  $  [a, b] $
is said to be quasi-analytic in the sense of Hadamard if the equalities  $  f ^ { (n) } (c) = 0 $, 
$  n = 0, 1 \dots $
at some fixed point  $  c $, 
$  a < c < b $, 
imply that  $  f \equiv 0 $. 
The statement of the theorem: The class  $  K $
is quasi-analytic if and only if

$$ \tag{1 }
( M _ {n} (f  ) )  ^ {1/n}  <  A (f  ) a _ {n} ,\ \ 
n = 0, 1 \dots
$$

where

$$ 
M _ {n} (f  )  = \ 
\max _ {a \leq  x \leq  b } \ 
| f ^ { (n) } (x) |,
$$

$  A (f  ) $
is a constant, and the sequence  $  \{ a _ {n} \} $
satisfies one of the equivalent conditions:

$$ \tag{2 }
\int\limits _ { 1 } ^  \infty  

\frac{ \mathop{\rm ln}  T (r)  dr }{r  ^ {2} }
  = \ 
+ \infty ,
$$

$$ 
\sum _ {n = 1 } ^  \infty   \left ( \inf _ {k \geq  n }
 a _ {k}  ^ {1/k} \right )  ^ {-1}  =  + \infty ,
$$

where

$$ 
T (r)  = \ 
\sup _ {n \geq  1 } \ 

\frac{r  ^ {n} }{a _ {n} }
 .
$$

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  $  s _ {n} $, 
$  n = 0, 1 \dots $
satisfies the condition

$$ 
\sum _ {n = 1 } ^  \infty  
\left ( 
\frac{1}{s _ {2n} }
 \right )  ^ {1/2n}
 =  + \infty ,
$$

then the moment problem

$$ \tag{3 }
s _ {k}  = \ 
\int\limits _ {- \infty } ^  \infty  
t  ^ {k}  d \sigma (t),\ \ 
k = 0, 1 \dots
$$

is well-defined. This means that there exists a non-decreasing function  $  \sigma (t) $, 
$  - \infty < t < + \infty $, 
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  $  f (x) $
is any continuous function on the real line and  $  \epsilon (r) $, 
$  0 < r < + \infty $, 
is a positive continuous function decreasing arbitrarily rapidly as  $  r \rightarrow + \infty $, 
then there exists an entire function  $  g (z) $
of the complex variable  $  z = x + iy $
such that

$$ 
| f (x) - g (x) |  < \ 
\epsilon ( | x | ),\ \ 
- \infty < x < + \infty .
$$

This theorem, established by T. Carleman , was the starting point in the investigations into approximation by entire functions. In particular, a continuum  $  E $
in the  $  z $-
plane is said to be a Carleman continuum if for any continuous complex function  $  f (z) $
on  $  E $
and an arbitrary rapidly decreasing positive function  $  \epsilon (r) $(
as  $  r \rightarrow \infty $) 
with a positive infimum on any finite interval, there exists an entire function  $  g (z) $
such that

$$ 
| f (z) - g (z) |  < \ 
\epsilon ( | z | ),\ \ 
z \in E.
$$

Necessary and sufficient conditions for a closed set  $  E $
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

$$ 
 \mathop{\rm arg}  z  = \ 
\textrm{ const } ,\ \ 
| z | > c > 0.
$$

Carleman's theorem on the approximation of analytic functions by polynomials in the mean over the area of a domain: Let  $  D $
be a finite domain in the complex  $  z $-
plane,  $  z = x + iy $, 
bounded by a Jordan curve  $  \Gamma $, 
and let  $  f (z) $
be a regular analytic function in  $  D $
such that

$$ 
{\int\limits \int\limits } _ { D }
| f (z) |  ^ {p}  dx  dy  <  \infty ,\ \ 
p > 0.
$$

Then there exists for any  $  \epsilon > 0 $
a polynomial  $  P (z) $
such that

$$ 
{\int\limits \int\limits } _ { D }
| f (z) - P (z) |  ^ {p}  dx  dy  <  \epsilon .
$$

This result was established by T. Carleman [[#References|[4]]]. Similar results also hold for approximation with an arbitrary positive continuous weight, in which case the boundary  $  \Gamma $
can be of a more general nature. The system of monomials  $  \{ z  ^ {n} \} $, 
$  n = 0, 1 \dots $
is complete with respect to any such weight. Orthogonalization and normalization of this system gives polynomials  $  P _ {n} (z) $
of degree  $  n $, 
which are often called Carleman polynomials.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  T. Carleman,   "Les fonctions quasi-analytiques" , Gauthier-Villars  (1926)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  T. Carleman,   "Sur les équations intégrales singulières à noyau réel et symmétrique"  ''Univ. Årsskrift'' :  3 , Uppsala  (1923)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  T. Carleman,   "Sur un théorème de Weierstrass"  ''Arkiv. Mat. Astron. Fys.'' , '''20''' :  4  (1927)  pp. 1–5</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  T. Carleman,   "Über die Approximation analytischer Funktionen durch lineare Aggregate von vorgegebenen Potenzen"  ''Arkiv. Mat. Astron. Fys.'' , '''17''' :  9  (1922)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  S. Mandelbrojt,   "Séries adhérentes, régularisations des suites, applications" , Gauthier-Villars  (1952)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  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</TD></TR></table>

====Comments====
The following result is also known as Carleman's theorem. If  $  F (z) $
is a holomorphic function in the region

$$ 
G  =  \{ {z } : {
0 < \lambda \leq  | z | \leq  R,\ 
 \mathop{\rm Im}  z \geq  0 } \}
$$

and  $  a _ {k} = r _ {k} e ^ {i \theta _ {k} } $, 
$  k = 1 \dots n $, 
are the zeros of  $  F $(
counted with multiplicity) in  $  G $, 
then

$$ 
\sum _ {\lambda < r _ {k} < R }
\left ( {
\frac{1}{r _ {k} }
 } -

\frac{r _ {k} }{R  ^ {2} }

\right )  \sin  \theta _ {k}  = \ 
{
\frac{1}{\pi R }
 }
\int\limits _ { 0 } ^  \pi  
 \mathop{\rm ln}  | F ( R e ^ {i \theta } ) | \ 
\sin  \theta  d \theta +
$$

$$ 
+ 
{
\frac{1}{2 \pi }
 } \int\limits _  \lambda  ^ { R }  \left ( {
\frac{1}{x
 ^ {2} }
 } - {
\frac{1}{R  ^ {2} }
 } \right )   \mathop{\rm ln} \ 
| F (x) F (-x) |  dx + A _  \lambda  (F, R),
$$

where

$$ 
A _  \lambda  (F, R)  = \ 
-  \mathop{\rm Im}  {
\frac{1}{2 \pi }
 }
\int\limits _ { 0 } ^  \pi  
 \mathop{\rm ln}  F ( \lambda e ^ {i \theta } )
\left (

\frac{\lambda e ^ {i \theta } }{R  ^ {2} }
 -
{
\frac{e ^ {-i \theta } } \lambda 
 }
\right )  d \theta .
$$

See [[#References|[a2]]]. Further, [[#References|[a1]]] is a good reference for the approximation theorems in the present article.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  D. Gaier,   "Vorlesungen über Approximation im Komplexen" , Birkhäuser  (1980)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  B.Ya. Levin,   "Distribution of zeros of entire functions" , Amer. Math. Soc.  (1980)  (Translated from Russian)</TD></TR></table>