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
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.
[1]: 226 In 1930, Jovan Karamata gave a new and much simpler proof.
[1]: 226 This formulation is from Titchmarsh.
we have The theorem is sometimes quoted in equivalent forms, where instead of requiring
[2]: 155 The theorem is sometimes quoted in another equivalent formulation (through the change of variable
[2]: 155 If, then The following more general formulation is from Feller.
[3]: 445 Consider a real-valued function
is defined by the Stieltjes integral The theorem relates the asymptotics of ω with those of
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
to be a piecewise constant function with value
According to the definition of a slowly varying function,
is slow varying at infinity iff for every
be a function slowly varying at infinity and
Then the following statements are equivalent Karamata (1930) found a short proof of the theorem by considering the functions
such that An easy calculation shows that all monomials
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
The theorem can fail without the condition that the coefficients are non-negative.
For example, the function is asymptotic to
, but the partial sums of its coefficients are 1, 0, 2, 0, 3, 0, 4, ... and are not asymptotic to any linear function.
In 1911 Littlewood proved an extension of Tauber's converse of Abel's theorem.
, 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 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.