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The Littlewood Tauberian Theorem Cambridge University Press 978-1-107-05945-0 - Twelve Landmarks of Twentieth-Century Analysis D. Choimet and H. Queffélec Excerpt More information 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. 1 © in this web service Cambridge University Press www.cambridge.org Cambridge University Press 978-1-107-05945-0 - Twelve Landmarks of Twentieth-Century Analysis D. Choimet and H. Queffélec Excerpt More information 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. © in this web service Cambridge University Press www.cambridge.org Cambridge University Press 978-1-107-05945-0 - Twelve Landmarks of Twentieth-Century Analysis D. Choimet and H. Queffélec Excerpt More information 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 © in this web service Cambridge University Press www.cambridge.org Cambridge University Press 978-1-107-05945-0 - Twelve Landmarks of Twentieth-Century Analysis D. Choimet and H. Queffélec Excerpt More information 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. © in this web service Cambridge University Press www.cambridge.org Cambridge University Press 978-1-107-05945-0 - Twelve Landmarks of Twentieth-Century Analysis D. Choimet and H. Queffélec Excerpt More information 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). © in this web service Cambridge University Press www.cambridge.org Cambridge University Press 978-1-107-05945-0 - Twelve Landmarks of Twentieth-Century Analysis D. Choimet and H. Queffélec Excerpt More information 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 hypo- thesis nan → 0bynan bounded, which Hardy had done the year before using the method of Cesàro summation (see Theorem 1.2.6).
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