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In [[mathematical analysis]], the '''Hardy–Littlewood tauberian theorem''' is a [[tauberian theorem]] relating the [[asymptotics]] of the partial sums of a [[series (mathematics)|series]] with the asymptotics of its [[Abel summation]].  In this form, the theorem asserts that if, as ''y'' ↓ 0, the sequence ''a''<sub>''n''</sub> is such that there is an [[asymptotic equivalence]]
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:<math>\sum_{n=0}^\infty a_n e^{-ny} \sim \frac{1}{y}</math>
then there is also an asymptotic equivalence
:<math>\sum_{k=0}^n a_k \sim n</math>
as ''n'' &rarr; &infin;.  The [[integral]] formulation of the theorem relates in an analogous manner the asymptotics of the [[cumulative distribution function]] of a function with the asymptotics of its Laplace transform.


The theorem was proved in 1914 by [[G. H. Hardy]] and [[J. E. Littlewood]].<ref name=Titchmarsh>
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{{cite book |last1=Titchmarsh |first1=E. C. |authorlink1=Edward Charles Titchmarsh |title=The Theory of Functions |edition=2nd |year=1939 |publisher=Oxford University Press |location=Oxford |isbn=0-19-853349-7}}
</ref>{{Rp|226}} In 1930 [[Jovan Karamata]] gave a new and much simpler proof.<ref name=Titchmarsh />{{Rp|226}}
 
==Statement of the theorem==
 
===Series formulation===
This formulation is from Titchmarsh.<ref name=Titchmarsh />{{Rp|226}} Suppose ''a''<sub>''n''</sub> ≥ 0 for all ''n'', and as ''x'' ↑1 we have
:<math>\sum_{n=0}^\infty a_n x^n \sim \frac{1}{1-x}.</math>
Then as ''n'' goes to ∞ we have
:<math>\sum_{k=0}^n a_k \sim n.</math>
The theorem is sometimes quoted in equivalent forms, where instead of requiring ''a''<sub>''n''</sub> ≥ 0, we require ''a''<sub>''n''</sub> = O(1), or we require ''a''<sub>''n''</sub> ≥ &minus;''K'' for some constant ''K''.<ref name=Hardy>
{{cite book |last1=Hardy |first1=G. H. |authorlink1=G. H. Hardy |title=Divergent Series |year=1991 |origyear=1949 |publisher=AMS Chelsea |location=Providence, RI |isbn=0-8284-0334-1}}
</ref>{{Rp|155}} The theorem is sometimes quoted in another equivalent formulation (through the change of variable ''x'' = 1/''e''<sup>''y''</sup> ).<ref name=Hardy />{{Rp|155}} If, as ''y'' ↓ 0,
:<math>\sum_{n=0}^\infty a_n e^{-ny} \sim \frac{1}{y}</math>
then
:<math>\sum_{k=0}^n a_k \sim n.</math>
 
===Integral formulation===
The following more general formulation is from Feller.<ref>{{Cite book | last1=Feller | first1=William | author1-link=William Feller | title=An introduction to probability theory and its applications. Vol. II. | publisher=[[John Wiley & Sons]] | location=New York | series=Second edition | id={{MathSciNet | id = 0270403}} | year=1971}}
</ref>{{Rp|445}} Consider a real-valued function ''F''&nbsp;:&nbsp;[0,∞)&nbsp;→&nbsp;'''R''' of [[bounded variation]].<ref>Bounded variation is only required locally: on every bounded subinterval of [0,∞).  However, then more complicated additional assumptions on the convergence of the Laplace–Stieltjes transform are required. See {{Cite book | last1=Shubin | first1=M. A. | title=Pseudodifferential operators and spectral theory | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Springer Series in Soviet Mathematics | isbn=978-3-540-13621-7 | id={{MathSciNet | id = 883081}} | year=1987}}</ref>  The [[Laplace–Stieltjes transform]] of ''F'' is defined by the [[Stieltjes integral]]
 
:<math>\omega(s) = \int_0^\infty e^{-st}\,dF(t).</math>
 
The theorem relates the asymptotics of ω with those of ''F'' in the following way.  If ρ is a non-negative real number, then the following are equivalent
:<math>\omega(s)\sim C s^{-\rho},\quad\rm{as\ }s\to 0</math>
:<math>F(t)\sim \frac{C}{\Gamma(\rho+1)}t^\rho, \quad\rm{as\ }t\to\infty.</math>
Here Γ denotes the [[Gamma function]]. One obtains the theorem for series as a special case by taking ρ&nbsp;=&nbsp;1 and ''F''(''t'') to be a piecewise constant function with value <math>\textstyle{\sum_{k=0}^n a_k}</math> between ''t''=''n'' and ''t''=''n''+1.
 
A slight improvement is possible. A function ''L''(''x'') is '''slowly varying at infinity''' if
:<math>\frac{L(tx)}{L(x)}\to 1,\quad x\to\infty</math>
for every positive ''t''.  Let ''L'' be a function slowly varying at infinity and ρ a non-negative real number. Then the following are equivalent
:<math>\omega(s)\sim s^{-\rho}L(s^{-1}),\quad\rm{as\ }s\to 0</math>
:<math>F(t)\sim \frac{1}{\Gamma(\rho+1)}t^\rho L(t), \quad\rm{as\ }t\to\infty.</math>
 
==Examples==
 
===Littlewood's extension of Tauber's theorem===
In 1911 [[John Edensor Littlewood|Littlewood]] proved an extension of [[Alfred Tauber|Tauber]]'s converse of [[Abel's theorem]]. Littlewood showed the following: If ''a''<sub>''n''</sub> = O(1/''n'' ), and as ''x'' ↑ 1 we have
:<math>\sum a_n x^n \to s, </math>
then
:<math> \sum a_n = s.</math>
This came historically before the Hardy–Littlewood tauberian theorem, but can be proved as a simple application of it.<ref name=Titchmarsh />{{Rp|233–235}}
 
===Prime number theorem===
In 1915 Hardy and Littlewood developed a proof of the [[prime number theorem]] based on their tauberian theorem; they proved
:<math>\sum_{n=2}^\infty \Lambda(n) e^{-ny} \sim \frac{1}{y},</math>
where Λ is the [[von Mangoldt function]], and then conclude
:<math> \sum_{n \le x} \Lambda(n) \sim x,</math>
an equivalent form of the prime number theorem.<ref name=12Ramanujan>
{{cite book | last = Hardy | first = G. H. |authorlink=G. H. Hardy | title = Ramanujan: Twelve Lectures on Subjects Suggested by his Life and Work | publisher = AMS Chelsea Publishing | location = Providence | year = 1999 |origyear=1940 | isbn = 978-0-8218-2023-0 }}</ref>{{Rp|34–35}}<ref name=Narkiewicz>
{{cite book | last = Narkiewicz | first = Władysław | title = The Development of Prime Number Theory | publisher = Springer-Verlag | location = Berlin | year = 2000 | isbn = 3-540-66289-8 }}
</ref>{{Rp|302–307}}
Littlewood developed a simpler proof, still based on this tauberian theorem, in 1971.<ref name=Narkiewicz />{{Rp|307–309}}
 
==Notes==
{{reflist}}
 
==External links==
* {{springer|id=t/t092280|title=Tauberian theorems}}
* {{MathWorld |title=Hardy-Littlewood Tauberian Theorem |urlname=Hardy-LittlewoodTauberianTheorem}}
 
{{DEFAULTSORT:Hardy-Littlewood tauberian theorem}}
[[Category:Tauberian theorems]]
[[Category:Theorems in analysis]]

Latest revision as of 10:48, 1 May 2014

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