The Littlewood Tauberian Theorem

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1 The Littlewood Tauberian theorem

1.1 Introduction

In 1897, the Austrian mathematician Alfred Tauber published a short article on the convergence of numerical series [173], which can be summarised as follows. ∞ = Let an be a convergent series of complex numbers, with n=0 an .A theorem of Abel [1] states that ∞ n f (x) = an x → as x 1. (1.1) n=0

A theorem of Kronecker [116] states that

n 1 → . n kak 0 (1.2) k=1

The converse of these two theorems is false: neither of the two conditions (1) nor (2) is sufficient to imply the convergence of the series an. However, if both conditions are satisfied simultaneously, the series an converges, giving the following theorem. 1.1.1 Theorem A necessary and sufficient condition for an to converge (with sum ) is that:

∞ (1) f (x) = a xn → as x 1, n=0 n 1 n → (2) n k=1 kak 0.

The proof of Theorem 1.1.1 follows that of the following special case.

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2 The Littlewood Tauberian theorem

( ) → → ∞ = 1.1.2 Theorem f x as x 1, and nan 0 implies n=0 an . A few remarks on this article: the theorem of Abel cited above gave rise to Abelian theorems [113], that is, theorems of the form If an is a convergent complex series with sum , and (bn,x ) an inﬁnite rectangular matrix indexed by N × X, where X is a set with an associated point at inﬁnity satisfying

bn,x −→ 1 forevery n ∈ N, x→∞ then ∞ f (x) = anbn,x n=0 is deﬁned for x ∈ X and

f (x) → as x →∞.

Such a theorem generalises the theorem of Abel, where we have X =[0, 1[, = n the point at infinity being 1, and bn,x x . It also generalises the case n + b , = 1 − and X = N, the point at infinity being ∞, which n x x + 1 corresponds to ∞ S0 +···+Sx a b , = , n n x x + 1 n=0 where Sn is the partial sum of index n of the series an. The corresponding Abelian theorem is none other than the theorem of Cauchy–Cesàro. The theorem of Kronecker is now referred to as the lemma of Kronecker [153]. For the proof of Theorem 1.1.2, we proceed as follows. With the inequality 1 − xn n(1 − x) if 0 x < 1, (1.3) − − 1 → = +···+ we show that Sn f 1 n 0, where Sn a0 an. It is essentially Theorem 1.1.2 which has passed on to posterity. Despite its elegance, it remains relatively superficial, because of the highly restrictive hypothesis (nan → 0) and because of the limited use of the other hypo- − 1 → ( ) → thesis: we make use of f 1 n while in fact we have f x as x 1. Nonetheless, at least two jewels of theorems can be considered as direct descendants of Tauber’s theorem: the following results due to L. Fejér and A. Zygmund.

1.1 Introduction 3

1.1.3 Theorem [Fejér] Let D be the unit open disk, J a Jordan curve with interior , and f : D → a conformal mapping that can be extended to a homomorphism (also denoted) ffromD over = ∪ J. Then the Taylor series of f converges uniformly on D. ( ) = ∞ n The proof begins by showing that if f z n=0 an z , then the area of ∞ ∞ is π n|a |2. We thus know that n|a |2 < ∞, which implies n=1 n n=1 n 1 n a Tauberian-style condition = k|ak|→0. Next, setting Sn(θ) = n k 1 n ijθ = − 1 j=0 a j e and rn 1 n , we use Tauber’s method to show that iθ Sn(θ) − f (rne ) → 0 uniformly with respect to θ.

1.1.4 Theorem [Zygmund [195]] We consider the trigonometric series ∞ ( + ) n=1 an cos nx bn sin nx satisfying a Tauberian condition

N 1 lim nρn = 0, with ρn =|an|+|bn|. (1.4) N→∞ N n=1 − ( ) = ∞ an sin nx bn cos nx Then, the primitive series F x n=1 n converges normally on R. Moreover, setting N = 1 , we have h |h|

Nh F(x + h) − F(x) − (a cos nx + b sin nx) → 0 when |h|0, h n n n=1

uniformly with respect to x. The hypothesis (1.4) is veriﬁed for a lacunary ∞ (α + β ) series k=1 k cos nk x k sin nk x , with

nk+1 q > 1 and |αk|+|βk|→0. nk

In other words, under the hypothesis (1.4), we obtain a point-by-point result of differentiation term by term: the derivative series of F converges at a point x0 with sum if and only if the function F is differentiable at x0,

and in this case F (x0) = . For example, the real-valued function F(x) = −k ∞ 2√ ( k ) k=1 sin 2 x is almost everywhere non-differentiable, because the k k ∞ ( x) cos√2 lacunary series k=1 , where the squares of the coefﬁcients are k not summable, is almost everywhere divergent (see [15], Vol. 2, p. 242) and

4 The Littlewood Tauberian theorem

√1 → 0. However, F is differentiable on a non-countable set of points k because it belongs to the little Zygmund class (see Exercise 7.4).1 Here is the proof: deﬁne

n − = ρ ( ) = an sin nx bn cos nx Tn j j and Fn x n . j=1

A ﬁrst Abel transformation gives

N ρ N − n = Tn Tn−1 n 2 n=1 n=1 n N−1 T − − = N + T (n 2 − (n + 1) 2) 2 n N n=1 N−1 − N 1 + n = O(1), 3 n=1 n

where the notation A B means that A λB, where λ is a positive constant. This proves normal convergence. Moreover, Taylor provides the estimate

F (x + h) − F (x) n n = a cos nx + b sin nx + O(nρ |h|), (1.5) h n n n with O being uniform with respect to all the parameters. From this,

Nh F(x + h) − F(x) − (a cos nx + b sin nx) h n n n=1 Nh F (x + h) − F (x) F (x + h) − F (x) = n n − F (x) + n n , h n h n=1 n>Nh

1 We even have 1 F(x + h) − F(x) = O h log when h 0, h

uniformly with respect to x, which is better than the general estimate 1 G(x + h) − G(x) = o h ln h

for G in the little Zygmund class.

1.1 Introduction 5

ρ n from which, using (1.5) and the fact that Fn ∞ n : Nh F(x + h) − F(x) − (an cos nx + bn sin nx) h n=1 Nh ρ |h| nρ + 1 n . (1.6) n |h| n n=1 n>Nh

A further Abel transformation gives

ρ ρ − n = n n = Tn Tn−1 n n2 n2 n>Nh n>Nh n>Nh −2 −2 TNh = Tn(n − (n + 1) ) − , (N + 1)2 n>Nh h

so that2 ⎛ ⎞ ρ n = ⎝ n ⎠ + ( −1) = ( −1). o o Nh o Nh n n3 n>Nh n>Nh

Finally, referring back to (1.6): Nh Nh F(x + h) − F(x) 1 − (an cos nx + bn sin nx) |h| nρn + o h |h|N = = h n 1 n 1 1 = o(|h|Nh) + o |h|Nh = o(1),

with o being uniform. The lacunary case corresponds to an = αk when n = nk, an = 0 otherwise, and similarly for bn.Letε>0 and N 1 be ﬁxed, then p and k0 be indices 1 such that n p N < n p+1 and

γk ε for k k0, where γk =|αk|+|βk|.

2 This time, we use the full force of the hypothesis Tn = o(n).

6 The Littlewood Tauberian theorem

Then,3 N p nρn = nkγk n=1 k=1 − k0 1 p = nkγk + nkγk k=1 k=k0 − k0 1 −1 −2 nkγk + εn p 1 + q + q +··· k=1 −1 −1 Cε + Nε 1 − q and hence N −1 −1 −1 lim N nρn ε 1 − q for every ε>0, N→∞ n=1 so that N 1 nρ → 0 N n n=1 as stated. The aim of this chapter is to analyse in detail the enormous progress realised by Littlewood in 1911, when, in Tauber’s Theorem 1.1.2, he replaced the hypothesis nan → 0bynan bounded, which Hardy had done the year before using the method of Cesàro summation (see Theorem 1.2.6). Littlewood’s proof is nonetheless incredibly more elaborate than that of Hardy, and one can wonder why. In fact, Exercise 1.8 provides an indication: supposing that f (x) has a limit when x 1isaprioria much weaker supposition than that made by Hardy. Furthermore, when Tauber proved his two theorems, obviously he did not consider them as conditional converses of Abelian theorems, that is to say, theorems of the form

If ∞ f (x) = anbn,x → as x →∞ n=0

3 We use the bound, valid for 1 k p:

nk+1 nk+2 n p n ... . k q − q2 q p k

1.2 State of the art in 1911 7

( ) and if as well an n0 veriﬁes certain additional conditions, of smallness or 4 lacunarity for example, then an converges with sum .

The study of such converse theorems is precisely what Hardy and Little- wood undertook after 1911. They proposed naming them Tauberian theorems, in honour of Tauber and his seminal result. Tauber’s result would today be considered as a remark, by the standards of twentieth-century publications, which does not preclude a certain sense of depth and technical difﬁculty, as evidenced by the papers of Tauber and others. The subject of power series of a single variable is often considered a little old-fashioned, with the single issue of determining the radius of convergence (for which there are a number of rules, each as boring as the others). In Section 1.4 we see that it is in fact much more rich and complex, as soon as we approach the circle of convergence. Littlewood’s theorem already bears witness to this, and the subject poses problems that are open to this day! We start by examining the state of the art in 1911, before Littlewood’s paper, considered as the starting point of his 30-year collaboration with G. H. Hardy.

1.2 State of the art in 1911

In the following, we will consistently use the deﬁnitions below: ∞ ∞ ∞ n n 2 n f (x) = an x = (1 − x) Sn x = (1 − x) σn x , n=0 n=0 n=0

where Sn = a0 +···+an and σn = S0 +···+Sn. We suppose (without loss of generality) that our power series have radius of convergence 1 (or in any case 1). Before Littlewood, the principal results relating the behaviour of Sn, σn and f (x) when x 1 were as follows. σ → n → 1.2.1 Theorem [Cauchy mean (1821)] If Sn , then n . The n converse is false (Sn = u , |u|=1,u = 1).

1.2.2 Theorem [Abel continuity (1826)] If Sn → , then f (x) → . The n converse is false (an = u , |u|=1,u = 1). +···+ σ S0 Sn = n → 1.2.3 Theorem [Frobenius continuity (1880)] If n n , n then f (x) → . The converse is false (Sn = nu , |u|=1,u = 1). 4 1 n → For Tauber, this corresponds to the condition n k=0 kak 0.

8 The Littlewood Tauberian theorem

This result is more general than that of Abel because of the Cauchy mean theorem. As for Cesàro, he obtained the following results, which extend those of Cauchy (and in the process reprove a theorem of Mertens).

1.2.4 Theorem [Cesàro multiplication (1890)] (1) If an → a and bn → b, 1 ( +···+ ) → then n a0bn anb0 ab. (2) If an and bn converge respectively to A and B, and if cn = +···+ = + ··· + C0 Cn → i+ j=n ai b j and Cn c0 cn, then n AB. In particular, if Cn → C, we necessarily have C = AB (theorem of Mertens).

Statement (1) is in fact a simple improvement of Theorem 1.2.1. As we have already mentioned, Tauber [113, 124] had already established the following conditional converse of Abel’s theorem.

( ) → → 1.2.5 Theorem [Tauber (1897)] If f x and nan 0(or if only 1 n → → n j=1 jaj 0), then Sn .

This theorem, the ﬁrst in a long line, is a gem (“remarkable”, according to Littlewood) even if somewhat superﬁcial. It is the ancestor of Fejér’s theorem, which can also be shown with the simple identity:

n n−1 1 ijθ = (θ) − 1 (θ), n jaj e Sn n S j j=1 j=0 (θ) = n ikθ where Sn k=0 ake . The second term on the righthand side con- f (eiθ ) verges uniformly to , after another theorem of Fejér. The ﬁrst term is 1 n | | bounded in absolute value by n j=1 j a j , hence it tends uniformly to 0. iθ As a result, Sn(θ) → f (e ) uniformly, without the necessity of invoking a Tauberian-style argument. However, it is difﬁcult to give a convincing example of an application of Theorem 1.2.5 (possibly the convergence of the series (−1)n| sin n| ). In the same vein we have the following two results. n ln n σ n → 1.2.6 Theorem [Hardy (1910)] If n , and in addition we suppose nan bounded, then Sn → .

1.2 State of the art in 1911 9

While it could still be considered “elementary” this theorem is nonetheless more difﬁcult, and possibly more useful, than that of Tauber, as it has applications to Fourier series (functions of bounded variation). Naturally, Hardy posed the question whether this converse remained true for the method of Abel, that is if, in Tauber’s theorem, we can replace nan → 0bynan bounded. We shall see Littlewood’s response later. Landau, in 1910, gave an improvement to Hardy’s theorem:σ for an real, the one-sided condition − n → → nan C is sufﬁcient to improve n to Sn .

1.2.7 Theorem [Fatou (1905) [113, 157]] Suppose that f admits an ana- lytic continuation in the neighbourhood of z = 1, and that an → 0. Then, Sn → f (1).

This theorem was completely satisfactory, because it was clearly optimal: if ∞ a = un, with |u|=1 and u = 1, f (z) = (uz)n = 1 can be n n=0 1 − uz analytically continued around 1, however Sn diverges. This was not the case with the theorem of Tauber (or even that of Hardy). Keeping in mind the eternal n counter-example seen previously (an = u ), one could imagine a proposition of the form

if f (x) → and an → 0, then Sn → .

We will see later Littlewood’s answer (in the negative) to this question. Another subject in the air in 1911 was the more general case of the Dirichlet −λ x series ane n , with λn +∞(in fact, this is Littlewood’s framework) λ = and the case of the power series corresponding to n n via the change of −ε n −nε variable x = e : an x = ane . For these series, Landau proved the following generalisation of Tauber’s theorem.

1.2.8 Theorem [Landau (1907)] Set μn = λn − λn−1 > 0.Iff(ε) = ∞ μ −λnε → ε = n n=0 ane as 0, and in addition an o , then λn Sn = a0 +···+an → .

Up to a change in notation, the proof is the same as for the theorem of Tauber: 1 SN − f → 0, using the inequality λN λn −λnε −tε (λn − λn−1)e e dt. λn−1

10 The Littlewood Tauberian theorem

We now make a detailed analysis of Littlewood’s article and its contributions.

1.3 Analysis of Littlewood’s 1911 article

In 1911, John Edensor Littlewood was 26 years old: he was thus a beginner in the world of mathematics, as opposed to Godfrey Harold Hardy, who at 34 was already a confirmed and recognised mathematician. However, one can say that this article marks a triumphant entrance of the author into the big league. It contains (among others) four inspiring sections, that we will subsequently examine one by one. (1) An analysis of Tauber’s proof, and a non-trivial counter-example illus- trating to what extent the conditions nan → 0 and |nan| C are different. (2) The tour de force of the article: the affirmative answer to Hardy’s question, in a much more general context than that of power series – Dirichlet series. The proof was much more difficult than anything shown previously (see Exercise 1.8 for a heuristic justification). (3) A proof of the optimality of the result obtained, again in a very general framework, with (prophetic) estimations of independent interest of sums of imaginary exponentials. (4) A work plan, executed successfully afterwards, that gave birth to the field of Tauberian theorems for which, a century later, J. Korevaar [113] published his monumental Tauberian Theory, a Century of Developments.

1.3.1 A non-trivial example

When we analyse the proof of Tauber (nan → 0), we see that the essential point is as follows: SN − f (x) → 0ifN →+∞and x 1 in such a manner that C1 N(1 − x) C2, where C1 and C2 are positive constants. It follows that by designating F the cluster set in C ∪{∞}of f (x), when x 1, and S the cluster set of SN , when N →+∞,wehave:

if nan → 0, then F = S. (1.7) This statement is evidently a generalisation of Tauber’s theorem; later (Had- wiger et al.) it was shown that (at least in the case where f (x) and SN are bounded):

h(S, F) H lim |nan|, (1.8) n→+∞