- Carlson's theorem
In
mathematics , in the area ofcomplex analysis , Carlson's theorem is auniqueness theorem about a summable expansion of ananalytic function . It is typically invoked to defend the uniqueness of aNewton 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 themaximum-modulus theorem .tatement of theorem
If "f"("z") is an
entire function ofexponential type , in other words, if:f(z) = mathcal{O}(1) e^{ au|z
for some au < 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 the sine function.
A variation
Let f(z) be regular, of the form :mathcal{O}(e^{ au|z) for operatorname{Re}(z)ge 0, and let :f(iy)=mathcal{O}(e^{-a|y) for a>0, on the imaginary axis z=iy. Then f(z)=0 identically.
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 Asubset mathbb{N} which is sufficiently dense in mathbb{N}. That is, a sufficient condition is given by having the set "A" satisfy
:lim_{n oinfty} 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" 'thforward 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.
* L.A. Rubel, " [http://www.pubmedcentral.nih.gov/articlerender.fcgi?artid=528143 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.
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