Inmathematics, in the area ofcomplex analysis,Carlson's theorem is auniqueness theorem which was discovered byFritz David Carlson. Informally, it states that two differentanalytic functions which do not grow very fast at infinity can not coincide at theintegers. The theorem may be obtained from thePhragmén–Lindelöf theorem, which is itself an extension of themaximum-modulus theorem.
Carlson's theorem is typically invoked to defend the uniqueness of aNewton series expansion. Carlson's theorem has generalized analogues for other expansions.
Assume thatf satisfies the following three conditions. The first two conditions bound the growth off at infinity, whereas the third one states thatf vanishes on the non-negative integers.
Thenf is identically zero.
The first condition may be relaxed: it is enough to assume thatf is analytic inRez > 0,continuous inRez ≥ 0, and satisfiesfor some real valuesC,τ.
To see that the second condition is sharp, consider the functionf(z) = sin(πz). It vanishes on the integers; however, it grows exponentially on theimaginary axis with a growth rate ofc =π, and indeed it is not identically zero.
A result due toRubel (1956) relaxes the condition thatf vanish on the integers. Namely, Rubel showed that the conclusion of the theorem remains valid iff vanishes on a subsetA ⊂ {0, 1, 2, ...} ofupper density 1, meaning that
This condition is sharp, meaning that the theorem fails for setsA of upper density smaller than 1.
Supposef(z) is a function that possesses all finiteforward differences. Consider then theNewton serieswhere is thebinomial coefficient and is then-th forward difference. By construction, one then has thatf(k) =g(k) for all non-negative integersk, so that the differenceh(k) =f(k) −g(k) = 0. This is one of the conditions of Carlson's theorem; ifh obeys the others, thenh is identically zero, and the finite differences forf uniquely determine its Newton series. That is, if a Newton series forf exists, and the difference satisfies the Carlson conditions, thenf is unique.