Carlson's theorem

In mathematics, in the area of 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.

Statement of theorem

If f(z) is an entire function of exponential type, in other words, if

f(z) = \mathcal{O}(1) e^{\tau|z|}

for some $\tau < \infty$ , and O is big-O notation, and if

f(iy) = \mathcal{O}(1) e^{c|y|}

for some $c < \pi$

and if f vanishes identically on the non-negative integers, then f is identically zero.

As a counter-example, note that $\sin \pi z$ vanishes on the integers; however, it fails to satisfy the second condition (since it grows exponentially on the imaginary axis, with a growth rate of $c=\pi$ ), and so Carlson's theorem does not apply to sin.

Rubel extension

A result, due to L.A. Rubel, relaxes the condition that f vanish on the integers slightly, so that f need vanish only on a set

A\subset \mathbb{N}

which is sufficiently dense in

\mathbb{N}

. That is, a sufficient condition is given by having the set A satisfy

\lim_{n\to\infty} \frac{A(n)}{n} = 1

where A(n) is the number of integers in A that are less than n.

Alternative formulation

An alternative formulation, due to W. H. J. Fuchs, replaces the requirement that f be entire with the requirement that f be regular for Re z>1/2.

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.
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.
L.A. Rubel, "Necessary and Sufficient Conditions for Carlson's Theorem on Entire Functions", Proc Natl Acad Sci U S A. 1955 August 15; 41(8): 601–603.

Complex analysis | Factorial and binomial topics | Finite differences | Mathematical theorems

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Statement of theorem

Rubel extension

Alternative formulation

Applications

References