Английская Википедия:Carlson's theorem
Шаблон:Distinguish Шаблон:Short description In mathematics, in the area of complex analysis, Carlson's theorem is a uniqueness theorem which was discovered by Fritz David Carlson. Informally, it states that two different analytic functions which do not grow very fast at infinity can not coincide at the integers. The theorem may be obtained from the Phragmén–Lindelöf theorem, which is itself an extension of the maximum-modulus theorem.
Carlson's theorem is typically invoked to defend the uniqueness of a Newton series expansion. Carlson's theorem has generalized analogues for other expansions.
Statement
Assume that Шаблон:Math satisfies the following three conditions. The first two conditions bound the growth of Шаблон:Math at infinity, whereas the third one states that Шаблон:Math vanishes on the non-negative integers.
- Шаблон:Math is an entire function of exponential type, meaning that <math display="block">|f(z)| \leq C e^{\tau|z|}, \quad z \in \mathbb{C}</math> for some real values Шаблон:Math, Шаблон:Math.
- There exists Шаблон:Math such that <math display="block">|f(iy)| \leq C e^{c|y|}, \quad y \in \mathbb{R} </math>
- Шаблон:Math for every non-negative integer Шаблон:Math.
Then Шаблон:Math is identically zero.
Sharpness
First condition
The first condition may be relaxed: it is enough to assume that Шаблон:Math is analytic in Шаблон:Math, continuous in Шаблон:Math, and satisfies
<math display="block">|f(z)| \leq C e^{\tau|z|}, \quad \operatorname{Re} z > 0</math>
for some real values Шаблон:Math, Шаблон:Math.
Second condition
To see that the second condition is sharp, consider the function Шаблон:Math. It vanishes on the integers; however, it grows exponentially on the imaginary axis with a growth rate of Шаблон:Math, and indeed it is not identically zero.
Third condition
A result, due to Шаблон:Harvtxt, relaxes the condition that Шаблон:Math vanish on the integers. Namely, Rubel showed that the conclusion of the theorem remains valid if Шаблон:Math vanishes on a subset Шаблон:Math of upper density 1, meaning that
<math display="block"> \limsup_{n \to \infty} \frac{\left| A \cap \{0,1,\ldots,n-1\} \right|}{n} = 1. </math>
This condition is sharp, meaning that the theorem fails for sets Шаблон:Math of upper density smaller than 1.
Applications
Suppose Шаблон:Math is a function that possesses all finite forward differences <math>\Delta^n f(0)</math>. Consider then the Newton series
<math display="block">g(z) = \sum_{n=0}^\infty {z \choose n} \, \Delta^n f(0)</math>
with <math display="inline">{z \choose n}</math> is the binomial coefficient and <math>\Delta^n f(0)</math> is the Шаблон:Math-th forward difference. By construction, one then has that Шаблон:Math for all non-negative integers Шаблон:Math, so that the difference Шаблон:Math. This is one of the conditions of Carlson's theorem; if Шаблон:Math obeys the others, then Шаблон:Math is identically zero, and the finite differences for Шаблон:Math uniquely determine its Newton series. That is, if a Newton series for Шаблон:Math exists, and the difference satisfies the Carlson conditions, then Шаблон:Math is unique.
See also
References
- F. Carlson, Sur une classe de séries de Taylor, (1914) Dissertation, Uppsala, Sweden, 1914.
- Шаблон:Cite journal, cor 21(1921) p. 6.
- Шаблон:Cite journal
- E.C. Titchmarsh, The Theory of Functions (2nd Ed) (1939) Oxford University Press (See section 5.81)
- R. P. Boas, Jr., Entire functions, (1954) Academic Press, New York.
- Шаблон:Cite journal
- Шаблон:Cite journal
- Шаблон:Citation
