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.
Carlson's theorem is typically invoked to defend the uniqueness of a Newton series expansion.
The first two conditions bound the growth of f at infinity, whereas the third one states that f vanishes on the non-negative integers.
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
It vanishes on the integers; however, it grows exponentially on the imaginary axis with a growth rate of c = π, and indeed it is not identically zero.
A result, due to Rubel (1956), relaxes the condition that f vanish on the integers.
Namely, Rubel showed that the conclusion of the theorem remains valid if f vanishes on a subset A ⊂ {0, 1, 2, ...} of upper density 1, meaning that
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 forward differences
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.