In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, in the area of complex analysis

Complex analysis

Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is useful in many branches of mathematics, including number theory and applied mathematics; as well as in physics,...

, Carlson's theorem is a uniqueness theorem

Uniqueness theorem

The uniqueness theorem for Poisson's equation states that the equation has a unique gradient of the solution for a large class of boundary conditions...

 about a summable expansion of an analytic function

Analytic function

In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions, categories that are similar in some ways, but different in others...

. It is typically invoked to defend the uniqueness of a Newton series expansion. Carlson's theorem has generalized analogues for expansions in other bases of polynomials. It is named in honour of Fritz David Carlson

Fritz Carlson

Fritz David Carlson was a Swedish mathematician. He's famous for* Carlson's theorem in complex analysis* Carlson's inequality*Carlson–Levin constant*Carlson theorem on Dirichlet series...

.

The theorem may be obtained from the Phragmén–Lindelöf theorem, which is itself an extension of the maximum-modulus theorem.

Statement of theorem

Assume that
is an entire function

Entire function

In complex analysis, an entire function, also called an integral function, is a complex-valued function that is holomorphic over the whole complex plane...

 of exponential type, meaning that

for some

• There exists   such that

vanishes identically on the non-negative integers.

Then is identically zero.

First condition

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


for some

Second condition

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

Third condition

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


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

Applications

Suppose


is a Newton series, so that is the binomial coefficient

Binomial coefficient

In mathematics, binomial coefficients are a family of positive integers that occur as coefficients in the binomial theorem. They are indexed by two nonnegative integers; the binomial coefficient indexed by n and k is usually written \tbinom nk , and it is the coefficient of the x k term in...

 and is the n 'th forward difference. Carlson's theorem then states that if all vanish, then is identically zero. As a trivial corollary, if a Newton series for f exists, and satisfies the Carlson conditions, then f is unique.