# Hardy–Littlewood Tauberian theorem

Hardy–Littlewood Tauberian theorem (Redirected from Hardy–Littlewood tauberian theorem) Jump to navigation Jump to search 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 {displaystyle sum _{n=0}^{infty }a_{n}e^{-ny}sim {frac {1}{y}}} then there is also an asymptotic equivalence {displaystyle sum _{k=0}^{n}a_{k}sim n} 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 Contents 1 Statement of the theorem 1.1 Series formulation 1.2 Integral formulation 2 Karamata's proof 3 Examples 3.1 Non-positive coefficients 3.2 Littlewood's extension of Tauber's theorem 3.3 Prime number theorem 4 Notes 5 External links Statement of the theorem Series formulation This formulation is from Titchmarsh.[1]: 226 Suppose an ≥ 0 for all n, and as x ↑ 1 we have {displaystyle sum _{n=0}^{infty }a_{n}x^{n}sim {frac {1}{1-x}}.} Then as n goes to ∞ we have {displaystyle sum _{k=0}^{n}a_{k}sim n.} 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, {displaystyle sum _{n=0}^{infty }a_{n}e^{-ny}sim {frac {1}{y}}} then {displaystyle sum _{k=0}^{n}a_{k}sim n.} Integral formulation 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 {displaystyle omega (s)=int _{0}^{infty }e^{-st},dF(t).} 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 {displaystyle omega (s)sim Cs^{-rho },quad {rm {{as }sto 0}}} {displaystyle F(t)sim {frac {C}{Gamma (rho +1)}}t^{rho },quad {rm {{as }tto infty .}}} 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 {displaystyle textstyle {sum _{k=0}^{n}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 {displaystyle {frac {L(tx)}{L(x)}}to 1,quad xto infty } 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 {displaystyle omega (s)sim s^{-rho }L(s^{-1}),quad {rm {{as }sto 0}}} {displaystyle F(t)sim {frac {1}{Gamma (rho +1)}}t^{rho }L(t),quad {rm {{as }tto infty .}}} Karamata's proof Karamata (1930) found a short proof of the theorem by considering the functions g such that {displaystyle lim _{xrightarrow 1}(1-x)sum a_{n}x^{n}g(x^{n})=int _{0}^{1}g(t)dt} 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. Examples Non-positive coefficients The theorem can fail without the condition that the coefficients are non-negative. For example, the function {displaystyle {frac {1}{(1+x)^{2}(1-x)}}=1-x+2x^{2}-2x^{3}+3x^{4}-3x^{5}+cdots } 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 theorem 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 an = O(1/n), and as x ↑ 1 we have {displaystyle sum a_{n}x^{n}to s,} then {displaystyle sum a_{n}=s.} 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 In 1915 Hardy and Littlewood developed a proof of the prime number theorem based on their Tauberian theorem; they proved {displaystyle sum _{n=2}^{infty }Lambda (n)e^{-ny}sim {frac {1}{y}},} where Λ is the von Mangoldt function, and then conclude {displaystyle sum _{nleq x}Lambda (n)sim x,} 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 ^ Jump up to: a b c d Titchmarsh, E. C. (1939). The Theory of Functions (2nd ed.). Oxford: Oxford University Press. ISBN 0-19-853349-7. ^ Jump up to: a b Hardy, G. H. (1991) [1949]. Divergent Series. Providence, RI: AMS Chelsea. ISBN 0-8284-0334-1. ^ Feller, William (1971). An introduction to probability theory and its applications. Vol. II. Second edition. New York: John Wiley & Sons. MR 0270403. ^ 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. ^ 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. ^ Jump up to: a b Narkiewicz, Władysław (2000). The Development of Prime Number Theory. Berlin: Springer-Verlag. ISBN 3-540-66289-8. External links "Tauberian theorems", Encyclopedia of Mathematics, EMS Press, 2001 [1994] Weisstein, Eric W. "Hardy-Littlewood Tauberian Theorem". MathWorld. Categories: Tauberian theorems

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