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, as y ↓ 0, the non-negative sequence an is such that there is an asymptotic equivalence

then there is also an asymptotic equivalence

as n → ∞. 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 theoremEdit

Series formulationEdit

This formulation is from Titchmarsh.[1]:226 Suppose an ≥ 0 for all n, and as x ↑1 we have

 

Then as n goes to ∞ we have

 

The theorem is sometimes quoted in equivalent forms, where instead of requiring an ≥ 0, we require an = O(1), or we require an ≥ −K for some constant K.[2]:155 The theorem is sometimes quoted in another equivalent formulation (through the change of variable x = 1/ey ).[2]:155 If, as y ↓ 0,

 

then

 

Integral formulationEdit

The following more general formulation is from Feller.[3]:445 Consider a real-valued function F : [0,∞) → R of bounded variation.[4] The Laplace–Stieltjes transform of F is defined by the Stieltjes integral

 

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 statements are equivalent

  •  
  •  

Here Γ denotes the Gamma function. One obtains the theorem for series as a special case by taking ρ = 1 and F(t) to be a piecewise constant function with value   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

 

for every positive t. Let L be a function slowly varying at infinity and ρ a non-negative real number. Then the following statements are equivalent

  •  
  •  

Karamata's proofEdit

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

 

An easy calculation shows that all monomials g(x)=xk 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 an 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 Σanxng(xn) is a0+...+aN, and the integral of g is 1, from which the Hardy–Littlewood theorem follows immediately.

ExamplesEdit

Non-positive coefficientsEdit

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

 

is asymptotic to 1/4(1–x) as x tends to 1, 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 theoremEdit

In 1911 Littlewood proved an extension of Tauber's converse of Abel's theorem. Littlewood showed the following: If an = O(1/n), and as x ↑ 1 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 theoremEdit

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

 

where Λ 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

NotesEdit

  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 [0,∞). 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.

External linksEdit