# 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 $K$ 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

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