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

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)