Tauberian theoremsFrom Wikipedia, the free encyclopediaContents1 Abelian and tauberian theorems 11.1 Abelian theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Tauberian theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.4 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 HardyLittlewood tauberian theorem 32.1 Statement of the theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.1.1 Series formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.1.2 Integral formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2 Karamatas proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.3.1 Non-positive coecients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.3.2 Littlewoods extension of Taubers theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 52.3.3 Prime number theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.5 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 Littlewoods Tauberian theorem 73.1 Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73.2 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 Slowly varying function 94.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 Wieners tauberian theorem 115.1 The condition in L1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115.1.1 Tauberian reformulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115.1.2 Discrete version . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12iii CONTENTS5.2 The condition in L2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125.3 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125.5 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 WienerIkehara theorem 146.1 Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146.2 Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156.4 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 166.4.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166.4.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166.4.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16Chapter 1Abelian and tauberian theoremsIn mathematics, abelianandtauberiantheorems are theorems giving conditions for two methods of summingdivergent series to give the same result, named after Niels Henrik Abel and Alfred Tauber. The original examplesare Abels theorem showing that if a series converges to some limit then its Abel sum is the same limit, and Tauberstheorem showing that if the Abel sum of a series exists and the coecients are suciently small (o(1/n)) then theseries converges to the Abel sum. More general abelian and tauberian theorems give similar results for more generalsummation methods.There is no clear distinction between abelian and tauberian theorems, or even a generally accepted denition of whatthese terms mean. Often, a theorem is called abelian if it shows that some summation method gives the usual sumfor convergent series, and is called tauberian if it gives conditions for a series summable by some method to besummable in the usual sense.1.1 Abelian theoremsFor any summation method L, its abelian theorem is the result that if c = (cn) is a convergent sequence, with limit C,then L(c) = C. An example is given by the Cesro method, in which L is dened as the limit of the arithmetic meansof the rst N terms of c, as N tends to innity. One can prove that if c does converge to C, then so does the sequence(dN) wheredN=c1 + c2 + + cNN.To see that, subtract C everywhere to reduce to the case C = 0. Then divide the sequence into an initial segment,and a tail of small terms: given any > 0 we can take N large enough to make the initial segment of terms up to cNaverage to at most /2, while each term in the tail is bounded by /2 so that the average is also necessarily bounded.The name derives from Abels theorem on power series. In that case L is the radial limit (thought of within thecomplex unit disk), where we let r tend to the limit 1 from below along the real axis in the power series with termanznand set z = rei. That theorem has its main interest in the case that the power series has radius of convergence exactly1: if the radius of convergence is greater than one, the convergence of the power series is uniform for r in [0,1] sothat the sum is automatically continuous and it follows directly that the limit as r tends up to 1 is simply the sum ofthe an. When the radius is 1 the power series will have some singularity on |z| = 1; the assertion is that, nonetheless,if the sum of the an exists, it is equal to the limit over r. This therefore ts exactly into the abstract picture.1.2 Tauberian theoremsPartial converses to abelian theorems are called tauberian theorems. The original result of Tauber (1897) statedthat if we assume also12 CHAPTER 1. ABELIAN AND TAUBERIAN THEOREMSan = o(1/n)(see Little o notation) and the radial limit exists, then the series obtained by setting z = 1 is actually convergent.This was strengthened by J. E. Littlewood: we need only assume O(1/n).A sweeping generalization is the HardyLittlewood tauberian theorem.In the abstract setting, therefore, an abelian theorem states that the domain of L contains convergent sequences, andits values there are equal to those of the Lim functional. A tauberian theorem states, under some growth condition,that the domain of L is exactly the convergent sequences and no more.If one thinks of L as some generalised type of weighted average, taken to the limit, a tauberian theorem allows oneto discard the weighting, under the correct hypotheses. There are many applications of this kind of result in numbertheory, in particular in handling Dirichlet series.The development of the eld of tauberian theorems received a fresh turn with Norbert Wiener's very general results,namely Wieners tauberian theorem and its large collection of corollaries. The central theorem can now be proved byBanach algebra methods, and contains much, though not all, of the previous theory.1.3 ReferencesKorevaar, Jacob (2004). Tauberian theory. A century of developments. Grundlehren der MathematischenWissenschaften 329. Springer-Verlag. ISBN 978-3-540-21058-0.Montgomery, Hugh L.; Vaughan, Robert C. (2007). Multiplicative number theory I. Classical theory. Cam-bridge tracts in advanced mathematics 97. Cambridge: Cambridge Univ. Press. pp. 147167. ISBN 0-521-84903-9.Tauber, A. (1897). Ein Satz aus der Theorie der unendlichen Reihen (A theorem from the theory of inniteseries)". Monatsh. F. Math. (in German) 8: 273277. doi:10.1007/BF01696278. JFM 28.0221.02.Wiener, N. (1932). Tauberian theorems. Annals of Mathematics33 (1): 1100. doi:10.2307/1968102.JSTOR 1968102.1.4 External linksHazewinkel, Michiel, ed. (2001), Tauberian theorems, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4Chapter 2HardyLittlewood tauberian theoremIn mathematical analysis, the HardyLittlewood tauberian theorem is a tauberian theorem relating the asymptoticsof 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 equivalencen=0aneny1ythen there is also an asymptotic equivalencenk=0ak nas n . The integral formulation of the theorem relates in an analogous manner the asymptotics of the cumulativedistribution 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 andmuch simpler proof.[1]:2262.1 Statement of the theorem2.1.1 Series formulationThis formulation is from Titchmarsh.[1]:226 Suppose an 0 for all n, and as x 1 we haven=0anxn11 x.Then as n goes to we havenk=0ak n.The theorem is sometimes quoted in equivalent forms, where instead of requiring an 0, we require an = O(1), orwe 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 0,34 CHAPTER 2. HARDYLITTLEWOOD TAUBERIAN THEOREMn=0aneny1ythennk=0ak n.2.1.2 Integral formulationThe following more general formulation is from Feller.[3]:445 Consider a real-valued function F : [0,) R ofbounded variation.[4] The LaplaceStieltjes transform of F is dened by the Stieltjes integral(s) =0estdF(t).The theorem relates the asymptotics of with those of F in the following way. If is a non-negative real number,then the following are equivalent(s) Cs, as s 0F(t) C( + 1)t, as t .Here denotes the Gamma function. One obtains the theorem for series as a special case by taking = 1 and F(t) tobe a piecewise constant function with valuenk=0 ak between t=n and t=n+1.A slight improvement is possible. A function L(x) is slowly varying at innity ifL(tx)L(x) 1, x for every positive t. Let L be a function slowly varying at innity and a non-negative real number. Then the followingare equivalent(s) sL(s1), as s 0F(t) 1( + 1)tL(t), as t .2.2 Karamatas proofKaramata (1930) found a short proof of the theorem by considering the functions g such thatlimx1(1 x)anxng(xn) =10g(t)dtAn easy calculation shows that all monomials g(x)=xkhave this property, and therefore so do all polynomials g. Thiscan be extended to a function g with simple (step) discontinuities by approximating it by polynomials from above andbelow (using the Weierstrass approximation theorem and a little extra fudging) and using the fact that the coecientsan are positive.In particular the function given by g(t)=1/t if 1/e