Carlson's theorem

Template: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 Template:Math satisfies the following three conditions. The first two conditions bound the growth of Template:Math at infinity, whereas the third one states that Template:Math vanishes on the non-negative integers.

1. Template:Math is an entire function of exponential type, meaning that $$|f(z)|\le C{e}^{\tau |z|},z\in \mathbb{ℂ}$$ for some real values Template:Math, Template:Math.
2. There exists Template:Math such that $$|f(iy)|\le C{e}^{c|y|},y\in \mathbb{ℝ}$$
3. Template:Math for every non-negative integer Template:Math.

Then Template:Math is identically zero.

Sharpness

First condition

The first condition may be relaxed: it is enough to assume that Template:Math is analytic in Template:Math, continuous in Template:Math, and satisfies

$$|f(z)|\le C{e}^{\tau |z|},\mathrm{Re}z>0$$

for some real values Template:Math, Template:Math.

Second condition

To see that the second condition is sharp, consider the function Template:Math. It vanishes on the integers; however, it grows exponentially on the imaginary axis with a growth rate of Template:Math, and indeed it is not identically zero.

Third condition

A result, due to Template:Harvtxt, relaxes the condition that Template:Math vanish on the integers. Namely, Rubel showed that the conclusion of the theorem remains valid if Template:Math vanishes on a subset Template:Math of upper density 1, meaning that

$\underset{n\to \mathrm{\infty }}{\mathrm{lim\; sup}}$ $\frac{|A\cap \{0,1,\dots ,n-1\}|}{n}=1.$

This condition is sharp, meaning that the theorem fails for sets Template:Math of upper density smaller than 1.

Applications

Suppose Template:Math is a function that possesses all finite forward differences ${\mathrm{\Delta }}^{n}f(0)$. Consider then the Newton series

$$g(z)=\sum _{n=0}^{\mathrm{\infty }}(\genfrac{}{}{0ex}{}{z}{n}){\mathrm{\Delta }}^{n}f(0)$$

where $(\genfrac{}{}{0ex}{}{z}{n})$ is the binomial coefficient and ${\mathrm{\Delta }}^{n}f(0)$ is the Template:Math-th forward difference. By construction, one then has that Template:Math for all non-negative integers Template:Math, so that the difference Template:Math. This is one of the conditions of Carlson's theorem; if Template:Math obeys the others, then Template:Math is identically zero, and the finite differences for Template:Math uniquely determine its Newton series. That is, if a Newton series for Template:Math exists, and the difference satisfies the Carlson conditions, then Template:Math is unique.

See also

• Newton series
• Mahler's theorem
• Table of Newtonian series

References

• F. Carlson, Sur une classe de séries de Taylor, (1914) Dissertation, Uppsala, Sweden, 1914.
• , cor 21(1921) p. 6.
• 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.