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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]]
:<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>
{{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==
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===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]]

Revision as of 12:13, 16 February 2014


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Just ensure that you hire experienced Word - Press CMS developer who is experienced enough to perform the task of Word - Press customization to get optimum benefits of Word - Press CMS. The only problem with most is that they only offer a monthly plan, you never own the software and you can’t even install the software on your site, you must go to another website to manage your list and edit your autoresponder. I hope this short Plugin Dynamo Review will assist you to differentiate whether Plugin Dynamo is Scam or a Genuine. If you have any questions regarding exactly where and how to use wordpress backup plugin, you can get hold of us at the web-site. You or your web designer can customize it as per your specific needs. Socrates: (link to ) Originally developed for affiliate marketers, I've used this theme again and again to develop full-fledged web sites that include static pages, squeeze pages, and a blog.

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