Carlson's theorem

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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.

1 Statement of theorem
2 A variation
3 Rubel extension
4 Alternative formulation
5 Applications
6 References

[edit] 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 $c < π$

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

As a counter-example, note that $sin(π z)$ 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 $c = π$ ), and so Carlson's theorem does not apply to the sine function.

[edit] A variation

Let $f (z)$ be regular, of the form

$\mathcal{O}(e^{\tau|z|})$

for $\operatorname{Re}(z)\ge 0$ , and let

$f(iy)=\mathcal{O}(e^{-a|y|})$

for $a > 0$ , on the imaginary axis $z = i y$ . Then $f (z) = 0$ identically.

[edit] 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.

[edit] 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.

[edit] 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 $Δ n f (0)$ is the n 'th forward difference. Carlson's theorem then states that if all $Δ n f (0)$ 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.

[edit] 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.
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.