Carlson's theorem

In

.

Carlson's theorem is typically invoked to defend the uniqueness of a

expansion. Carlson's theorem has generalized analogues for other expansions.

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

1. f(z) is an entire function of exponential type, meaning that $${\displaystyle |f(z)|\leq Ce^{\tau |z|},\quad z\in \mathbb {C} }$$ for some real values C, τ.
2. There exists c < π such that $${\displaystyle |f(iy)|\leq Ce^{c|y|},\quad y\in \mathbb {R} }$$
3. f(n) = 0 for every non-negative integer n.

Then f is identically zero.

The first condition may be relaxed: it is enough to assume that f is analytic in Re z > 0, continuous in Re z ≥ 0, and satisfies

$${\displaystyle |f(z)|\leq Ce^{\tau |z|},\quad \operatorname {Re} z>0}$$

for some real values C, τ.

To see that the second condition is sharp, consider the function f(z) = sin(πz). It vanishes on the integers; however, it grows exponentially on the

with a growth rate of c = π, and indeed it is not identically zero.

A result, due to

1, meaning that

${\displaystyle \limsup _{n\to \infty }}$ ${\displaystyle {\frac {\left|A\cap \{0,1,\ldots ,n-1\}\right|}{n}}=1.}$

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

Suppose f(z) is a function that possesses all finite

${\displaystyle \Delta ^{n}f(0)}$. Consider then the

$${\displaystyle g(z)=\sum _{n=0}^{\infty }{z \choose n}\,\Delta ^{n}f(0)}$$

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

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

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• E.C. Titchmarsh
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  doi:10.1090/s0002-9947-1962-0141920-6
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  PMID 16578453