Carlson's theorem

Not to be confused with Carleson's theorem

In mathematics, in the area of complex analysis, Carlson's theorem is a uniqueness theorem about a summable expansion of an analytic function. 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.

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

$f (z)$ is an entire function of exponential type, meaning that

|f(z)|\leq Ce^{\tau |z|},\quad z\in \mathbb {C}

for some

C, τ < \infty

There exists $c <$ $π$ such that

|f(iy)|\leq Ce^{c|y|},\quad y\in \mathbb {R}

$f$ vanishes identically on the non-negative integers.

Then $f$ is identically zero.

Sharpness

First condition

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

|f(z)|\leq Ce^{\tau |z|},\quad \Im z\geq 0

for some $C, τ < \infty$

Second condition

To see that the second condition is sharp, consider the function $f (z)$ = $sin$ ( $π$ $z$ ). 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.

Third condition

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 $A \subset {0,1,2,...$ } of upper density 1, meaning that

\limsup _{n\to \infty }{\frac {\#{\big (}A\cap \{0,1,\cdots ,n-1\}{\big )}}{n}}=1.

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

Applications

Suppose

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

is a Newton series, so that ${z \choose n}$ is the binomial coefficient and $\Delta ^{n}f(0)$ is the n 'th forward difference. Carlson's theorem then states that if all $\Delta ^{n}f(0)$ vanish, then $f(z)$ is identically zero. As a trivial corollary, if a Newton series for f exists, and satisfies the Carlson conditions, then f is unique.

References

F. Carlson, Sur une classe de séries de Taylor, (1914) Dissertation, Uppsala, Sweden, 1914.
M. Riesz, "Sur le principe de Phragmén–Lindelöf", Proceedings of the Cambridge Philosophical Society 20 (1920) 205–107, cor 21(1921) p.6.
G.H. Hardy, "On two theorems of F. Carlson and S. Wigert", Acta Mathematica, 42 (1920) 327–339.
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.
R. DeMar, "Existence of interpolating functions of exponential type", Trans. Amer. Math. Soc., 105 (1962) 359–371.
R. DeMar, "Vanishing Central Differences", Proc. Amer. math. Soc. 14 (1963) 64–67.
Rubel, L. A. (1956), "Necessary and sufficient conditions for Carlson's theorem on entire functions", Trans. Amer. Math. Soc., 83: 417–429, JSTOR 1992882, MR 0081944