Hardy–Littlewood Tauberian theorem explained

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] In 1930, Jovan Karamata gave a new and much simpler proof.

Statement of the theorem

Series formulation

This formulation is from Titchmarsh. 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] The theorem is sometimes quoted in another equivalent formulation (through the change of variable ). If, then

Integral formulation

The following more general formulation is from Feller.^[3] Consider a real-valued function

of bounded variation.^[4]

Notes and References

1. Book: Titchmarsh . E. C. . Edward Charles Titchmarsh . The Theory of Functions . 2nd . 1939 . Oxford University Press . Oxford . 0-19-853349-7.
2. Book: Hardy . G. H. . G. H. Hardy . Divergent Series . 1991 . 1949 . AMS Chelsea . Providence, RI . 0-8284-0334-1.
3. Book: Feller . William . William Feller . An introduction to probability theory and its applications. Vol. II. . . New York . Second edition . 0270403 . 1971.
4. Book: Hardy, G. H. . G. H. Hardy

   . G. H. Hardy . Ramanujan: Twelve Lectures on Subjects Suggested by his Life and Work . AMS Chelsea Publishing . Providence . 1999 . 1940 . 978-0-8218-2023-0 .

5. Book: Narkiewicz, Władysław . The Development of Prime Number Theory . Springer-Verlag . Berlin . 2000 . 3-540-66289-8 .
6. Bounded variation is only required locally: on every bounded subinterval of

   [0,infty)

   . However, then more complicated additional assumptions on the convergence of the Laplace–Stieltjes transform are required. See Book: Shubin . M. A. . Pseudodifferential operators and spectral theory . . Berlin, New York . Springer Series in Soviet Mathematics . 978-3-540-13621-7 . 883081 . 1987. The Laplace–Stieltjes transform of

   F

   is defined by the Stieltjes integral

   \omega(s)=

   	infty
   \int
   	0

   e^-stdF(t).

   The theorem relates the asymptotics of ω with those of

   F

   in the following way. If

   \rho

   is a non-negative real number, then the following statements are equivalent

   \omega(s)\simCs^-\rho,   \rm{as }s\to0

   F(t)\sim

   	C
   	\Gamma(\rho+1)

   t^\rho, as t\toinfty.

   Here

   \Gamma

   denotes the Gamma function. One obtains the theorem for series as a special case by taking

   \rho=1

   and

   F(t)

   to be a piecewise constant function with value

   	n
   style{\sum
   	k=0

   a_k}

   between

   t=n

   and

   t=n+1

   .

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

   L(x)

   is slow varying at infinity iff

   	L(tx)
   	L(x)

   \to1,   x\toinfty

   for every

   t>0

   . Let

   L

   be a function slowly varying at infinity and

   \rho\geq0

   . Then the following statements are equivalent

   \omega(s)\sims^-\rhoL(s^-1),   as s\to0

   F(t)\sim

   	1
   	\Gamma(\rho+1)

   t^\rhoL(t),   as t\toinfty.

   Karamata's proof

   found a short proof of the theorem by considering the functions

   g

   such that

   \lim_x\to(1-x)\sum

   	ng(x
   a
   	nx

   ⁿ⁾=

   	1
   \int
   	0

   g(t)dt

   An easy calculation shows that all monomials

   g(x)=x^k

   have this property, and therefore so do all polynomials

   g

   . This can be extended to a function

   g

   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

   a_n

   are positive. In particular the function given by

   g(t)=1/t

   if

   1/e<t<1

   and

   0

   otherwise has this property. But then for

   x=e^-1/N

   the sum

   \suma_nxⁿg(xⁿ⁾

   is

   a₀+ … +a_N

   and the integral of

   g

   is

   1

   , from which the Hardy–Littlewood theorem follows immediately.

   Examples

   Non-positive coefficients

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

   	1
   	(1+x)2(1-x)

   =1-x+2x^{2-2x3+3x4-3x5+ …}

   is asymptotic to

   1/4(1-x)

   as

   x\to1

   , 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

   See main article: Littlewood's Tauberian theorem. In 1911 Littlewood proved an extension of Tauber's converse of Abel's theorem. Littlewood showed the following: If

   a_n=O(1/n)

   , and we have

   \suma_nxⁿ\tos as x\uparrow1

   then

   \suma_n=s.

   This came historically before the Hardy–Littlewood Tauberian theorem, but can be proved as a simple application of it.

   Prime number theorem

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

   	infty
   \sum
   	n=2

   Λ(n)e^-ny\sim

   	1
   	y

   ,

   where

   Λ

   is the von Mangoldt function, and then conclude

   \sum_nΛ(n)\simx,

   an equivalent form of the prime number theorem.^{[4] [5]} Littlewood developed a simpler proof, still based on this Tauberian theorem, in 1971.

   Notes

   • Karamata. J.. Über die Hardy-Littlewoodschen Umkehrungen des Abelschen Stetigkeitssatzes. Mathematische Zeitschrift. December 1930. 32. 1. 319–320 . 10.1007/BF01194636. de.

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