Hardy–Littlewood Tauberian theorem

In mathematical analysis, the Hardy–Littlewood Tauberian theorem is a Tauberian theorem relating the asymptotics of the partial sums of a series with the asymptotics of its Abel summation. In this form, the theorem asserts that if the sequence is such that there is an asymptotic equivalence

then there is also an asymptotic equivalence

as . 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.[1]: 226  In 1930, Jovan Karamata gave a new and much simpler proof.[1]: 226 

Statement of the theorem

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Series formulation

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This formulation is from Titchmarsh.[1]: 226  Suppose   for all  , and we have

 

Then as   we have

 

The theorem is sometimes quoted in equivalent forms, where instead of requiring  , we require  , or we require   for some constant  .[2]: 155  The theorem is sometimes quoted in another equivalent formulation (through the change of variable  ).[2]: 155  If,

 

then

 

Integral formulation

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The following more general formulation is from Feller.[3]: 445  Consider a real-valued function   of bounded variation.[4] The Laplace–Stieltjes transform of   is defined by the Stieltjes integral

 

The theorem relates the asymptotics of ω with those of   in the following way. If   is a non-negative real number, then the following statements are equivalent

  •  
  •  

Here   denotes the Gamma function. One obtains the theorem for series as a special case by taking   and   to be a piecewise constant function with value   between   and  .

A slight improvement is possible. According to the definition of a slowly varying function,   is slow varying at infinity iff

 

for every  . Let   be a function slowly varying at infinity and  . Then the following statements are equivalent

  •  
  •  

Karamata's proof

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Karamata (1930) found a short proof of the theorem by considering the functions   such that

 

An easy calculation shows that all monomials   have this property, and therefore so do all polynomials  . This can be extended to a function   with simple (step) discontinuities by approximating it by polynomials from above and below (using the Weierstrass approximation theorem and a little extra fudging) and using the fact that the coefficients   are positive. In particular the function given by   if   and   otherwise has this property. But then for   the sum   is   and the integral of   is  , from which the Hardy–Littlewood theorem follows immediately.

Examples

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Non-positive coefficients

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The theorem can fail without the condition that the coefficients are non-negative. For example, the function

 

is asymptotic to   as  , but the partial sums of its coefficients are 1, 0, 2, 0, 3, 0, 4, ... and are not asymptotic to any linear function.

Littlewood's extension of Tauber's theorem

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In 1911 Littlewood proved an extension of Tauber's converse of Abel's theorem. Littlewood showed the following: If  , and we have

 

then

 

This came historically before the Hardy–Littlewood Tauberian theorem, but can be proved as a simple application of it.[1]: 233–235 

Prime number theorem

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In 1915 Hardy and Littlewood developed a proof of the prime number theorem based on their Tauberian theorem; they proved

 

where   is the von Mangoldt function, and then conclude

 

an equivalent form of the prime number theorem.[5]: 34–35 [6]: 302–307  Littlewood developed a simpler proof, still based on this Tauberian theorem, in 1971.[6]: 307–309 

Notes

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  1. ^ a b c d Titchmarsh, E. C. (1939). The Theory of Functions (2nd ed.). Oxford: Oxford University Press. ISBN 0-19-853349-7.
  2. ^ a b Hardy, G. H. (1991) [1949]. Divergent Series. Providence, RI: AMS Chelsea. ISBN 0-8284-0334-1.
  3. ^ Feller, William (1971). An introduction to probability theory and its applications. Vol. II. Second edition. New York: John Wiley & Sons. MR 0270403.
  4. ^ Bounded variation is only required locally: on every bounded subinterval of  . However, then more complicated additional assumptions on the convergence of the Laplace–Stieltjes transform are required. See Shubin, M. A. (1987). Pseudodifferential operators and spectral theory. Springer Series in Soviet Mathematics. Berlin, New York: Springer-Verlag. ISBN 978-3-540-13621-7. MR 0883081.
  5. ^ Hardy, G. H. (1999) [1940]. Ramanujan: Twelve Lectures on Subjects Suggested by his Life and Work. Providence: AMS Chelsea Publishing. ISBN 978-0-8218-2023-0.
  6. ^ a b Narkiewicz, Władysław (2000). The Development of Prime Number Theory. Berlin: Springer-Verlag. ISBN 3-540-66289-8.
  • Karamata, J. (December 1930). "Über die Hardy-Littlewoodschen Umkehrungen des Abelschen Stetigkeitssatzes". Mathematische Zeitschrift (in German). 32 (1): 319–320. doi:10.1007/BF01194636.
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