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| :''Not to be confused with [[Carleson's theorem]]''
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| In [[mathematics]], in the area of [[complex analysis]], '''Carlson's theorem''' is a [[uniqueness theorem]] which was discovered by [[Fritz Carlson|Fritz David Carlson]]. Informally, it states that two different analytic functions which do not grow very fast at infinity can not coincide at the integers. The theorem may be obtained from the [[Phragmén–Lindelöf theorem]], which is itself an extension of the [[maximum-modulus theorem]].
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| Carlson's theorem is typically invoked to defend the uniqueness of a [[Newton series]] expansion. Carlson's theorem has generalized analogues for other expansions.
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| ==Statement of theorem==
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| Assume that ''f'' satisfies the following three conditions: the first two conditions bound the growth of ''f'' at infinity, whereas the third one states that ''f'' vanishes on the non-negative integers.
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| * {{math|''f''(''z'')}} is an [[entire function]] of [[exponential type]], meaning that
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| ::<math>|f(z)| \leq C e^{\tau|z|}, \quad z \in \mathbb{C}</math>
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| :for some {{math|''C'', ''τ'' < ∞}}
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| * There exists {{math|''c'' <}} {{pi}} such that
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| ::<math>|f(iy)| \leq C e^{c|y|}, \quad y \in \mathbb{R} </math>
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| * {{math|''f''}}(''n'') = 0 for any non-negative integer ''n''.
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| Then {{math|''f''}} is identically zero.
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| ==Sharpness==
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| ===First condition===
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| The first condition may be relaxed: it is enough to assume that {{math|''f''}} is analytic in {{math|Re ''z'' > 0}}, continuous in {{math|Re ''z'' ≥ 0}}, and satisfies
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| :<math>|f(z)| \leq C e^{\tau|z|}, \quad \Im z \geq 0</math>
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| for some {{math|''C'',''τ'' < ∞}}
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| ===Second condition===
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| To see that the second condition is sharp, consider the function {{math|''f''(''z'')}} = {{math|sin}}({{pi}}{{math|''z''}}). It vanishes on the integers; however, it grows exponentially on the imaginary axis with a growth rate of {{math|''c''}} = {{pi}}, and indeed it is not identically zero.
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| ===Third condition===
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| A result, due to {{harvtxt|Rubel|1956}}, relaxes the condition that {{math|''f''}} vanish on the integers. Namely, Rubel showed that the conclusion of the theorem remains valid if {{math|''f''}} vanishes on a subset {{math|''A'' ⊂ {0,1,2,...}}} of [[upper density]] 1, meaning that
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| :<math> \limsup_{n \to \infty} \frac{\# \big( A \cap \{0,1,\cdots,n-1\} \big)}{n} = 1. </math> | |
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| This condition is sharp, meaning that the theorem fails for sets {{math|''A''}} of upper density smaller than 1.
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| ==Applications==
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| Suppose ''f''(''z'') is a function that possess all finite [[forward difference]]s <math>\Delta^n f(0)</math>. Consider then the [[Newton series]]
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| :<math>g(z)=\sum_{n=0}^\infty {z \choose n} \Delta^n f(0)</math>
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| with <math>{z \choose n}</math> is the [[binomial coefficient]] and <math>\Delta^n f(0)</math> is the ''n'' 'th [[forward difference]]. By construction, one then has that ''f''(''k'')=''g''(''k'') for all non-negative integers ''k'', so that the difference ''h''(''k'')=''f''(''k'')-''g''(''k'')=0. This is one of the conditions of Carlson's theorem; if ''h'' obeys the others, then ''h'' is identically zero, and the finite differences for ''f'' uniquely determine its Newton series. That is, if a Newton series for ''f'' exists, and the difference satisfies the Carlson conditions, then ''f'' is unique.
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| ==See also==
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| *[[Newton series]]
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| *[[Table of Newtonian series]]
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| ==References==
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| * F. Carlson, ''Sur une classe de séries de Taylor'', (1914) Dissertation, Uppsala, Sweden, 1914.
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| * [[M. Riesz]], "Sur le principe de Phragmén–Lindelöf", ''Proceedings of the Cambridge Philosophical Society'' '''20''' (1920) 205–107, cor '''21'''(1921) p.6.
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| * [[G.H. Hardy]], "On two theorems of F. Carlson and S. Wigert", ''Acta Mathematica'', '''42''' (1920) 327–339.
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| * [[E.C. Titchmarsh]], ''The Theory of Functions (2nd Ed)'' (1939) Oxford University Press ''(See section 5.81)''
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| * R. P. Boas, Jr., ''Entire functions'', (1954) Academic Press, New York.
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| * R. DeMar, "Existence of interpolating functions of exponential type", ''Trans. Amer. Math. Soc.'', '''105''' (1962) 359–371.
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| * R. DeMar, "Vanishing Central Differences", ''Proc. Amer. math. Soc. '''14''' (1963) 64–67.
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| * {{citation|mr=0081944|last=Rubel|first=L. A.|title=Necessary and sufficient conditions for Carlson's theorem on entire functions|journal=Trans. Amer. Math. Soc.|volume=83|year=1956|pages=417–429|jstor=1992882}}
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| [[Category:Factorial and binomial topics]]
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| [[Category:Finite differences]]
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| [[Category:Theorems in complex analysis]]
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