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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
 
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020430/c0204301.png" /> of real-valued infinitely differentiable functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020430/c0204302.png" /> on an interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020430/c0204303.png" /> is said to be quasi-analytic in the sense of Hadamard if the equalities <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020430/c0204304.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020430/c0204305.png" /> at some fixed point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020430/c0204306.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020430/c0204307.png" />, imply that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020430/c0204308.png" />. The statement of the theorem: The class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020430/c0204309.png" /> is quasi-analytic if and only if
+
(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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020430/c02043010.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
$$ \tag{1 }
 +
( M _ {n} (f  ) )  ^ {1/n}  < A (f  ) a _ {n} ,\ \
 +
n = 0, 1 \dots
 +
$$
  
 
where
 
where
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020430/c02043011.png" /></td> </tr></table>
+
$$
 +
M _ {n} (f  )  = \
 +
\max _ {a \leq  x \leq  b } \
 +
| f ^ { (n) } (x) |,
 +
$$
 +
 
 +
$  A (f  ) $
 +
is a constant, and the sequence  $  \{ a _ {n} \} $
 +
satisfies one of the equivalent conditions:
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020430/c02043012.png" /> is a constant, and the sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020430/c02043013.png" /> satisfies one of the equivalent conditions:
+
$$ \tag{2 }
 +
\int\limits _ { 1 } ^  \infty 
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020430/c02043014.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
\frac{ \mathop{\rm ln}  T (r)  dr }{r  ^ {2} }
 +
  = \
 +
+ \infty ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020430/c02043015.png" /></td> </tr></table>
+
$$
 +
\sum _ {n = 1 } ^  \infty  \left ( \inf _ {k \geq  n }
 +
a _ {k}  ^ {1/k} \right )  ^ {-1}  = + \infty ,
 +
$$
  
 
where
 
where
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020430/c02043016.png" /></td> </tr></table>
+
$$
 +
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.
 
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020430/c02043017.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020430/c02043018.png" /> satisfies the condition
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020430/c02043019.png" /></td> </tr></table>
+
$$
 +
\sum _ {n = 1 } ^  \infty 
 +
\left (
 +
\frac{1}{s _ {2n} }
 +
\right )  ^ {1/2n}
 +
= + \infty ,
 +
$$
  
 
then the moment problem
 
then the moment problem
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020430/c02043020.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>
+
$$ \tag{3 }
 +
s _ {k}  = \
 +
\int\limits _ {- \infty } ^  \infty 
 +
t  ^ {k}  d \sigma (t),\ \
 +
k = 0, 1 \dots
 +
$$
  
is well-defined. This means that there exists a non-decreasing function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020430/c02043021.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020430/c02043022.png" />, 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 , ).
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020430/c02043023.png" /> is any continuous function on the real line and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020430/c02043024.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020430/c02043025.png" />, is a positive continuous function decreasing arbitrarily rapidly as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020430/c02043026.png" />, then there exists an entire function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020430/c02043027.png" /> of the complex variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020430/c02043028.png" /> such that
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020430/c02043029.png" /></td> </tr></table>
+
$$
 +
| f (x) - g (x) |  < \
 +
\epsilon ( | x | ),\ \
 +
- \infty < x < + \infty .
 +
$$
  
This theorem, established by T. Carleman , was the starting point in the investigations into approximation by entire functions. In particular, a continuum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020430/c02043030.png" /> in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020430/c02043031.png" />-plane is said to be a Carleman continuum if for any continuous complex function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020430/c02043032.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020430/c02043033.png" /> and an arbitrary rapidly decreasing positive function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020430/c02043034.png" /> (as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020430/c02043035.png" />) with a positive infimum on any finite interval, there exists an entire function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020430/c02043036.png" /> such that
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020430/c02043037.png" /></td> </tr></table>
+
$$
 +
| f (z) - g (z) |  < \
 +
\epsilon ( | z | ),\ \
 +
z \in E.
 +
$$
  
Necessary and sufficient conditions for a closed set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020430/c02043038.png" /> 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
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020430/c02043039.png" /></td> </tr></table>
+
$$
 +
\mathop{\rm arg}  z  = \
 +
\textrm{ const } ,\ \
 +
| z | > c > 0.
 +
$$
  
Carleman's theorem on the approximation of analytic functions by polynomials in the mean over the area of a domain: Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020430/c02043040.png" /> be a finite domain in the complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020430/c02043041.png" />-plane, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020430/c02043042.png" />, bounded by a Jordan curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020430/c02043043.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020430/c02043044.png" /> be a regular analytic function in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020430/c02043045.png" /> such that
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020430/c02043046.png" /></td> </tr></table>
+
$$
 +
{\int\limits \int\limits } _ { D }
 +
| f (z) |  ^ {p}  dx  dy  < \infty ,\ \
 +
p > 0.
 +
$$
  
Then there exists for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020430/c02043047.png" /> a polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020430/c02043048.png" /> such that
+
Then there exists for any $  \epsilon > 0 $
 +
a polynomial $  P (z) $
 +
such that
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020430/c02043049.png" /></td> </tr></table>
+
$$
 +
{\int\limits \int\limits } _ { D }
 +
| f (z) - P (z) |  ^ {p}  dx  dy  < \epsilon .
 +
$$
  
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020430/c02043050.png" /> can be of a more general nature. The system of monomials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020430/c02043051.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020430/c02043052.png" /> is complete with respect to any such weight. Orthogonalization and normalization of this system gives polynomials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020430/c02043053.png" /> of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020430/c02043054.png" />, which are often called Carleman polynomials.
+
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====
 
====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>
 
<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
  
 +
$$
 +
G  =  \{ {z } : {
 +
0 < \lambda \leq  | z | \leq  R,\
 +
\mathop{\rm Im}  z \geq  0 } \}
 +
$$
  
====Comments====
+
and  $  a _ {k} = r _ {k} e ^ {i \theta _ {k} } $,
The following result is also known as Carleman's theorem. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020430/c02043055.png" /> is a holomorphic function in the region
+
$  k = 1 \dots n $,
 +
are the zeros of  $  F $(
 +
counted with multiplicity) in $  G $,
 +
then
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020430/c02043056.png" /></td> </tr></table>
+
$$
 +
\sum _ {\lambda < r _ {k} < R }
 +
\left ( {
 +
\frac{1}{r _ {k} }
 +
} -
  
and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020430/c02043057.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020430/c02043058.png" />, are the zeros of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020430/c02043059.png" /> (counted with multiplicity) in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020430/c02043060.png" />, then
+
\frac{r _ {k} }{R  ^ {2} }
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020430/c02043061.png" /></td> </tr></table>
+
\right )  \sin  \theta _ {k}  = \
 +
{
 +
\frac{1}{\pi R }
 +
}
 +
\int\limits _ { 0 } ^  \pi 
 +
\mathop{\rm ln}  | F ( R e ^ {i \theta } ) | \
 +
\sin  \theta  d \theta +
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020430/c02043062.png" /></td> </tr></table>
+
$$
 +
+
 +
{
 +
\frac{1}{2 \pi }
 +
} \int\limits _  \lambda  ^ { R }  \left ( {
 +
\frac{1}{x
 +
^ {2} }
 +
} - {
 +
\frac{1}{R  ^ {2} }
 +
} \right )  \mathop{\rm ln} \
 +
| F (x) F (-x) |  dx + A _  \lambda  (F, R),
 +
$$
  
 
where
 
where
  
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+
$$
 +
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.
 
See [[#References|[a2]]]. Further, [[#References|[a1]]] is a good reference for the approximation theorems in the present article.

Latest revision as of 10:23, 2 June 2020


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 [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

[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 $ 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 [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)
How to Cite This Entry:
Carleman theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Carleman_theorem&oldid=46218
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article