DISTRIBUTIONAL VERSIONS OF LITTLEWOOD’S TAUBERIAN THEOREM 2 RICARDOESTRADAANDJASSON VINDAS 1 0 2 Abstract. WeprovideseveralgeneralversionsofLittlewood’s Taube- rian theorem. These versions are applicable to Laplace transforms of n Schwartz distributions. We apply these Tauberian results to deduce a a J number of Tauberian theorems for power series where Ces`aro summa- bilityfollows fromAbelsummability. Wealsouseourgeneralresultsto 6 giveanewsimple proofof theclassical Littlewood one-sided Tauberian ] theorem for power series. A F . h t a 1. Introduction m [ A century ago, Littlewood obtained his celebrated extension of Tauber’s theorem [22, 14]. Littlewood’s Tauberian theorem states that if the se- 1 ∞ v ries n=0cn is Abel summable to the number a, namely, the power series 8 ∞ c rn has radius of convergence at least 1 and 0 n=P0 n 4 P ∞ 1 (1.1) lim rnc = a , n . r→1− 1 n=0 X 0 2 and if the Tauberian hypothesis 1 : 1 v (1.2) c = O n i n X (cid:18) (cid:19) ∞ ar is satisfied, then the series is actually convergent, n=0cn = a. The result was later strengthened by Hardy and Littlewood in [9, 10] to P an one-sided version. They showed that the condition (1.2) can be relaxed to the weaker one nc = O (1), i.e., there exists C > 0 such that n L −C <nc . n 2000 Mathematics Subject Classification. Primary 40E05, 40G10, 44A10, 46F12. Sec- ondary 40G05, 46F20. Keywordsandphrases. Tauberiantheorems;Laplacetransform;theconverseofAbel’s theorem; Littlewood’s Tauberian theorem; Abel and Ces`aro summability; distributional Tauberian theorems; asymptotic behavior of generalized functions. R.EstradagratefullyacknowledgessupportfromNSF,throughgrantnumber0968448. J.VindasgratefullyacknowledgessupportbyaPostdoctoralFellowshipoftheResearch Foundation–Flanders (FWO,Belgium). 1 2 RICARDOESTRADAANDJASSONVINDAS The aim of this article is to provide several distributional versions of this Hardy-Littlewood Tauberian theorem, our versions shall include it as a par- ticular case. Our general results are in terms of Laplace transforms of dis- tributions, and they have interesting consequences when applied to Stieltjes integralsandnumericalseries. Inparticular,weshallprovidevariousTaube- rian theorems where the conclusion is Cesa`ro (or Riesz) summability rather than convergence. We state a sample of our results. The ensuing theorem will be derived in Section 4.3 (cf. Corollary 4.4). In order to state it, we need to introduce some notation. We shall write b = O (1) (C,m) n L ∞ iftheCesa`romeansoforderm ≥ 1ofasequence{b } (nottobeconfused n n=0 withtheonesofaseries)areboundedfrombelow,namely,thereisaconstant K > 0 such that n m! k+m−1 −K < b . nm m−1 n−k k=0(cid:18) (cid:19) X ∞ Theorem 1.1. If c = a (A), then the Tauberian condition n=0 n (1.3) nc = O (1) (C,m) . P n L ∞ implies the (C,m) summability of the series, c = a (C,m). n=0 n TauberiantheoremsinwhichCesa`rosummaPbilityfollowsfromAbelsumma- bility have a long tradition, which goes back to Hardy and Littlewood [14, 11]. Such results have also received much attention in recent times, e.g., [1, 15]. Actually, Pati and C¸anak et al have made extensive use of Tauberian conditions involving the Cesa`ro means of nc , such as (1.3), in n the study of Tauberian theorems for the so called (A)(C,α) summability. We would like to point out that there is an extensive literature in Taube- rian theorems for Schwartz distributions, an overview can be found in [19, 27]. Extensions of the Wiener Tauberian theorem have been obtained in [16, 17, 18] (cf. [19]). Recent applications to the theory of Fourier and conjugate series are considered in [6]. We also mention that the results of this article are closely related to those from [7, 24], though with a different approach. For futurepurposes,itis convenient to restate Hardy-Littlewood theorem in a form which is invariant under addition of terms of the form n−1M. Set b = c , write b = c +C/n, for n > 0, and r = e−y. Then (1.1) transforms 0 0 n n into ∞ ∞ 1 b e−ny = −Clog(1−e−y)+ c e−ny = a+Clog +o(1) , n n y n=0 n=0 (cid:18) (cid:19) X X N whiletheconvergenceconclusiontranslatesinto b = a+Cγ+ClogN+ n=0 n o(1), N → ∞, where γ is the Euler gamma constant. Therefore, Hardy- P Littlewood theorem might be formulated as follows. DISTRIBUTIONAL VERSIONS OF LITTLEWOOD’S THEOREM 3 Theorem 1.2. Let ∞ c e−ny be convergent for y > 0. Suppose that n=0 n ∞ P 1 (1.4) lim c e−ny −blog = a . n y→0+ y n=0 (cid:18) (cid:19) X Then, the Tauberian hypothesis nc = O (1) implies that n L N (1.5) c = a+bγ+blogN +o(1) , N → ∞ . n n=0 X Theorem 1.2 is precisely the form of Littlewood’s theorem which we will generalize to distributions. The plan of this article is as follows. In Section 2 we explain the notions from distribution theory to be used in this paper. Section 3.3 provides a two-sided distributional version of Littlewood’s theo- rem. We shall use such a version to produce a simple proof of the classical Littlewood one-sided theorem. We give a one-sided Tauberian theorem for LaplacetransformsofdistributionsinSection 4andthendiscusssomeappli- cations to Stieltjes integrals and numerical series; as an example we extend a classical theorem of Sza´sz [21]. 2. Preliminaries and Notation 2.1. Distributions. The spaces of test functions and distributions D(R), S(R), D′(R), and S′(R) are well known for most analysts, we refer to [20, 26] for their properties. We denote by S[0,∞) the space of restrictions of test functions from S(R) to the interval [0,∞); its dual space S′[0,∞) is canonically isomorphic [26] to the subspace of distributions from S′(R) having supports in [0,∞). We shall employ several special distributions, we follow the notation ex- actly as in [5]. For instance, δ is as usual the Dirac delta, H is the Heaviside β−1 function,i.e., thecharacteristic functionof[0,∞), thedistributionsx are + simply given by xβ−1H(x) whenever ℜeβ > 0, and Pf(H(x)/x) is defined via Hadamard finite part regularization, i.e., H(x) 1 φ(x)−φ(0) ∞ φ(x) Pf ,φ(x) = dx+ dx . x x x (cid:28) (cid:18) (cid:19) (cid:29) Z0 Z1 2.2. Ces`aro Limits. We refer to [3, 5] for the Cesa`ro behavior of distribu- tions. We will only consider Cesa`ro limits. Given f ∈ D′(R) with support bounded at the left, we write (2.1) lim f(x)= ℓ (C,m) x→∞ if f(−m), the m-primitive of f with support bounded at the left, is an ordi- nary function for large arguments and ℓxm f(−m)(x) ∼ , x → ∞ . m! 4 RICARDOESTRADAANDJASSONVINDAS Observe that f(−m) is given by the convolution [26] xm−1 f(−m) = f ∗ + . (m−1)! If we do not want to make any reference to m in (2.1), we simply write (C). In the special case when f = s is a function of local bounded variation with s(x)= 0 for x < 0, then (2.1) reads as x t m lim 1− ds(t)= s(0)+ℓ . x→∞ x Z0 (cid:18) (cid:19) ∞ Thus, if s is given by the partial sums of a series c , this notion n=0 n ∞ amounts to the same as c = ℓ (C,m), as shown by the equivalence n=0 n P between Cesa`ro and Riesz summability [8, 12]. P 2.3. Laplace Transforms. Letf ∈ D′(R)besupportedin [0,∞), it is said to be Laplace transformable [20] on ℜez > 0 if e−yxf ∈ S′(R) is a tempered distributionforally > 0. InsuchacaseitsLaplace transformiswell defined on the half-plane ℜez > 0 and it is given by the evaluation L{f;z}= f(x),e−zx . If f = s is a function of local boun(cid:10)ded variatio(cid:11)n, then one readily verifies that it is Laplace transformable on ℜez > 0 in the distributional sense if and only if ∞ (2.2) L{ds;y}:= e−yxds(x) (C) exists for each y > 0 , Z0 Thus, Laplace transformability in this context is much more general than themereexistenceofLaplace-Stieltjes improperintegrals. Observealsothat the order of (C) summability might quickly change in (2.2) with each y. 2.4. Distributional Asymptotics. We shall make use of the theory of asymptotic expansions of distributions, explained for example in [5, 19, 23, 25]. For instance, let f,g ,g ∈ S′(R) and let c and c be two positive 1 2 1 2 functions such that c (λ) = o(c (λ)), λ → ∞. The asymptotic formula 2 1 f(λx)= c (λ)g (x)+c (λ)g (x)+o(c (λ)) as λ → ∞ in S′(R) , 1 1 2 2 2 is interpreted in the distributional sense, namely, it means that for all test functions φ∈ S(R) hf(λx),φ(x)i = c (λ)hg (x),φ(x)i+c (λ)hg (x),φ(x)i+o(c (λ)) . 1 1 2 2 2 3. Distributional Littlewood two-sided Tauberian Theorem We want to find a distributional analog to (1.5). Set s(x) = c , n<x n then (1.5) gives s(x) = a+bγ +blogx+o(1). It is now easy to prove [5, P Lem 3.9.2] that the previous ordinary expansion implies the distributional expansion s(λx) = (a+bγ)H(x)+bH(x)log(λx)+o(1) as λ → ∞ in S′(R) ; DISTRIBUTIONAL VERSIONS OF LITTLEWOOD’S THEOREM 5 differentiating [5], we obtain δ(x) b H(x) 1 s′(λx)= (a+bγ+blogλ) + Pf +o asλ → ∞ inS′(R). λ λ x λ (cid:18) (cid:19) (cid:18) (cid:19) The above distributional asymptotic relation is the one which we will mostly study in this article. In Subsection 3.1 we give an Abelian theorem related to it. We give a two-sided Tauberian converse in Subsection 3.3 that will be used to produce a new proof of Hardy-Littlewood theorem in the form of Theorem 1.2. The study of more general one-sided Tauberian conditions will be postponed to Section 4. 3.1. The Abelian Theorem. We begin with the following Abelian theo- rem for Laplace transforms of distributions. Theorem 3.1. Let g ∈ S′(R) be supported in [0,∞) and have the distribu- tional asymptotic behavior δ(x) logλ b H(x) (3.1) g(λx) = a +b δ(x)+ Pf +o(1) as λ → ∞ λ λ λ x (cid:18) (cid:19) in S′(R). Then, 1 (3.2) L{g;y} = a−bγ+blog +o(1) , y → 0+ . y (cid:18) (cid:19) Proof. Writing λ = y−1, we have, as λ → ∞, L g;λ−1 = λ g(λx),e−x H(x) (cid:8) (cid:9) = (a(cid:10)+blogλ) (cid:11)δ(x),e−x +b Pf ,e−x +o(1) x (cid:28) (cid:18) (cid:19) (cid:29) (cid:10) ∞(cid:11)e−x = a+blogλ+bF.p. dx+o(1) x Z0 = a+blogλ−bγ+o(1) . (cid:3) Corollary3.1. Letsbeafunctionoflocal bondedvariation suchthats(x) = 0 for x ≤ 0. If (3.3) lim (s(x)−blogx)= a (C) , x→∞ then, L{ds;y}:= ∞e−yxds(x) is (C) summable for each y > 0, and 0 R 1 (3.4) L{ds;y}= a−bγ +blog +o(1) , y → 0+ . y (cid:18) (cid:19) Proof. Set g = s′. The Cesa`ro limit (3.3) implies [5] that s(λx) = aH(x)+ bH(x)log(λx)+o(1) as λ → ∞ in S′(R). Differentiating, we conclude that g satisfies (3.1), and so, by Theorem 3.1, we deduce (3.4). (cid:3) 6 RICARDOESTRADAANDJASSONVINDAS In particular if we consider s(x) = c , we obtain that (1.5) implies n<x n (1.4), the Abelian counterpart of Theorem 1.2. P We end this subsection by pointing out that (3.1) is the most general asymptotic separation of variables we could have in the situation that we arestudying. Theproofofthefollowingpropositionfollowsfromthegeneral results from [4]. Proposition 3.1. Let g ∈ S′(R) be supported in [0,∞). If there are g ,g ∈ 1 2 S′(R) such that logλ 1 1 g(λx) = g (x)+ g (x)+o as λ → ∞ in S′(R) , 1 2 λ λ λ (cid:18) (cid:19) then g (x) = bδ(x) and g (x) = aδ(x)+bPf(H(x)/x), for some constants a 1 2 and b. Consequently, g has the distributional asymptotic behavior (3.1). 3.2. Functions and the Distributional Asymptotics (3.1). We shall provethat ifs isnon-decreasing ands′ has thedistributionalasymptotic be- havior (3.1), thenonerecovers theasymptotic behavior(3.3)intheordinary sense. Proposition 3.2. Let s ∈ L1 (R) be supported in [0,∞). If there exist loc A,B > 0 such that s(x)+Alogx is non-decreasing on the interval [B,∞) and δ(x) logλ b H(x) 1 (3.5) s′(λx) = a +b δ(x)+ Pf +o , λ λ λ x λ (cid:18) (cid:19) (cid:18) (cid:19) as λ → ∞ in S′(R), then (3.6) lim (s(x)−blogx) = a . x→∞ Proof. We may assume that s(0) = 0 and that s is non-decreasing on the whole R. Let ε be an arbitrary small number. Pick φ ,φ ∈ D(R) such that 1 2 0 ≤ φ ≤ 1, suppφ ⊆ [−1,1+ε], φ (x) = 1 for x ∈ [0,1], suppφ ⊆ [−1,1] j 2 2 1 and φ (x) = 1 for x ∈ [−1,1−ε]. Evaluating (3.5) at φ we have 1 2 ∞ x limsup(s(λ)−blogλ) ≤ lim φ ds(x)−blogλ 2 λ→∞ λ→∞(cid:18)Z0 (cid:16)λ(cid:17) (cid:19) ε+1 dx ε+1 dx = a+bF.p φ (x) = a+b φ (x) 2 2 x x Z0 Z1 ≤ a+bε . Likewise, evaluation at φ yields 1 ∞ dx 1 φ (x)−1 1 liminf(s(λ)−blogλ) ≥ a+bF.p φ (x) = a+b dx 1 λ→∞ Z0 x Z1−ε x ≥ a+blog(1−ε) . Since ε was arbitrary, we conclude (3.6). (cid:3) DISTRIBUTIONAL VERSIONS OF LITTLEWOOD’S THEOREM 7 3.3. Distributional two-sided Tauberian Theorem. We now show our firstdistributionalversionofLittlewoodTauberiantheorem. ItistheTaube- rian converse of Theorem 3.1. Since we use the big O symbol in the Taube- rian hypothesis, we denominate it a two-sided Tauberian theorem. Theorem 3.2. Let g ∈ S′(R) be supported on [0,∞). Suppose that, as y → 0+, 1 (3.7) L{g;y} = a+blog +o(1) . y (cid:18) (cid:19) Then, the Tauberian hypothesis δ(x) 1 (3.8) g(λx)−blogλ = O , λ λ (cid:18) (cid:19) implies the distributional asymptotic behavior δ(x) logλ b H(x) (3.9) g(λx) = (a+bγ) +b δ(x)+ Pf +o(1) . λ λ λ x (cid:18) (cid:19) Proof. Let g (x) = λg(λx) − blogλδ(x). Let B be the linear span of λ {e−τx} . Observe that B is dense in S[0,∞), due to the Hahn-Banach τ∈R theoremandthefactthattheLaplacetransformisinjective. Next, weverify that H(x) lim hg (x),φ(x)i = (a+bγ)δ(x)+bPf ,φ(x) , φ ∈B . λ λ→∞ x (cid:28) (cid:18) (cid:19) (cid:29) Indeed, it is enough for φ(x) = e−τx; by (3.7), as λ → ∞, τ λ g (x),e−τx = L g, −blogλ = a+blog −blogλ+o(1) λ λ τ (cid:18) (cid:19) n o (cid:10) (cid:11) H(x) = (a+bγ)δ(x)+bPf ,e−τx +o(1). x (cid:28) (cid:18) (cid:19) (cid:29) Now, the Tauberian hypothesis (3.8) implies that {g } is weakly λ λ∈[1,∞) boundedinS′[0,∞),andso,bytheBanach-Steinhaustheorem,itisequicon- tinuous. Sinceanequicontinuousfamilyoflinearfunctionalsconvergingover a dense subset must be convergent, we obtain that lim g (x) = (a+bγ)δ(x)+bPf(H(x)/x) in S′(R), λ λ→∞ which is precisely (3.9). (cid:3) 3.4. Classical Littlewood’s One-sided Theorem. Let us show how our two-sided Tauberian theorem can be used to give a simple proof of Hardy- Littlewood theorem in the form of Theorem 1.2. We actually give a more general result for Stieltjes integrals. Remark3.1. InmanyproofsofLittlewood’sone-sidedtheorem,suchasthe one based in Wiener’s method, one needs to establish first the boundedness of s(x) = c , which is not an easy task [8, 13, 28]. The method that n<x n P 8 RICARDOESTRADAANDJASSONVINDAS we develop in the proof of Theorem 3.3 rather estimates the second order Riesz means, which turns out to be much simpler. Theorem 3.3. Let s be of local bounded variation and supported in [0,∞). Suppose that (2.2) holds. Furthermore, assume that there exist A,B > 0 such that s(x)+Alogx is non-decreasing on [B,∞). Then 1 (3.10) L{ds;y} =a+blog +o(1) , y → 0+, y (cid:18) (cid:19) if and only if (3.11) s(x)= s(0)+a+bγ +blogx+o(1) , x → ∞ . Proof. One direction is implied by Corollary 3.1. For the other part, we may assume that s(0) = 0 and that s is non-decreasing over the whole real line. Consider the second order primitive s(−2)(x) = x(x−t)s(t)dt. Our 0 strategy will be to show R x2 (3.12) s(−2)(x) = b logx+O(x2) . 2 Supposeforthemomentthatwewereabletoshowthisclaim. Letusdeduce (3.11) from (3.12). By (3.12), we obtain the distributional relation (λx)2 s(−2)(λx) = b H(x)logλ+O(λ2) , 2 in S′(R). Differentiating three times, s′(λx)−bλ−1logλδ(x) = O(1/λ) in S′(R). ApplyingTheorem3.2tog = s′,weobtainthats′ hastheasymptotic behavior (3.9). Thus, Proposition 3.2 yields (3.12). Itthenremainstoshow(3.12). Westartbylookingats−1(x) = xs(t)dt. 0 Since 1−t ≤ e−t, we have the easy upper estimate R (3.13) s(−1)(x) = x 1− t ds(t)≤ ∞e−xtds(t)=blogx+O (1) . R x x Z0 (cid:18) (cid:19) Z0 Notice that (3.13) yields the upper estimate in (3.12). Next, define S(x) = bxlogx−s(−1)(x)+Cx, where the constant C > 0 is chosen so large that S(x) >0 for all x >0. Observe now that the lower estimate in (3.12) would immediately follow if we show x S(t)dt = O(x2) . Z0 Finally, because of (3.10), we have that ∞ lim y2 S(t)e−ytdt= b−γb−a+C , y→0+ Z0 and hence x x S(t)dt ≤e S(t)e−xtdt = O(x2) . Z0 Z0 The claim has been established and this completes the proof. (cid:3) DISTRIBUTIONAL VERSIONS OF LITTLEWOOD’S THEOREM 9 4. Littlewood One-sided Tauberian Theorems We want one-sided generalizations of Theorem 3.3 in which the conclu- sion is Cesa`ro limits. The generalization is in terms of Cesa`ro one-sided boundedness as explained in the next subsection. We shall show below first a Tauberian theorem for Laplace transforms of distributions. In Subsection 4.2 we study Stieltjes integrals and generalize a result of Sza´sz [21]. Finally, we give applications to numerical series in Subsection 4.3; in particular, we prove Theorem 1.1 . 4.1. Distributional Littlewood One-sided Tauberian Theorem. For the distributional generalization, let us rewrite the Tauberian hypothesis of Theorem 3.3 is a more suitable way for our purposes. Recall a distribution g is said to be non-negative on an interval (B ,B ) if it consides with a 1 2 non-negative measure on that interval. In such case we may write g(x) ≥ 0 on (B ,B ). With this notation the Tauberian hypothesis of Theorem 3.3 1 2 becomes s′(x)+A/x ≥ 0 on (B,∞), for some A,B > 0 or, multiplying by x, xs′(x) = O (1) on (B,∞). We can also generalize these ideas by using L the symbol O (1) in the Cesa`ro sense. L Definition 4.1. Let g ∈ D′(R). Given m ∈ N, we say that g(x) = O (1) (C,m) , x → ∞ , L if there exist A,B > 0 and a non-negative measure µ such that g(x)+A = µ(m) on (B,∞). Definition 4.1 makes possible to give sense to the relation xf′(x) = O (1) L in the Cesa`ro sense. We need also to introduce some notation in order to move further. For each m ∈ N, let l be the m-primitive of H(x)logx with support in [0,∞). m It can be verified by induction that for m ≥ 1 1 x xm xm m 1 (4.1) l (x) = log(t)(x−t)m−1dt = + logx− + . m (m−1)! m! m! k Z0 k=1 X Let f be supported on [0,∞). We now study the asymptotic behavior f(x)= a+blogx+o(1) (C,m), which in view of (4.1) means that xm xm m 1 (4.2) f(−m)(x)= b + logx+ + a−b +o(xm) , m! m! k ! k=1 X x → ∞, in the ordinary sense. The ensuing Tauberian theorem is a natural distributional version of Littlewood’s Tauberian theorem, in the context of Cesa`ro limits. Theorem 4.1. Let f ∈ D′(R) be such that suppf ⊆ [0,∞) and let m ∈ N. Assume that (4.3) xf′(x) = OL(1) (C,m) , x → ∞ . 10 RICARDOESTRADAANDJASSONVINDAS Suppose that f is Laplace transformable on ℜez = y > 0. Then, 1 (4.4) L f′,y = a+blog +o(1) , y → 0+ . y (cid:18) (cid:19) if and only if (cid:8) (cid:9) (4.5) lim f(x)−blogx = a+bγ (C,m) , x → ∞ . x→∞ Proof. We shall show that f(−m) is locally integrable for large arguments and xm f(−m)(x) = (a+bγ) +l (x)+o(xm) , x → ∞ . m m! Setting τ(x) = x−mf(−m), the above asymptotic formula is the same as m a bγ b 1 b (4.6) τ(x) = + − + logx+o(1) , x → ∞ . m! m! m! k m! k=1 X By adding a term of the form AH(x)logx to f and removing a compactly supported distribution, we may assume that (xf′)(−m) is a non-negative measure. It is clear that we can also assume that f, and hence f(−m), is zero in a neighborhood of the origin. Next, it is easy to verify that (xf′)(−m) = xf(−m+1) − mf(−m); multiplying by x−m−1, we obtain that τ′ = x−mf(−m+1)−mx−m−1f(−m) is a non-negative measure. We now look at the Laplace transform of τ′. Set F(y) = L f′;y and T(y) = L τ′;y . We then have, (cid:8) (cid:9) (cid:8) (cid:9) T(y) (m) dm ∞ f(−m)(x) ∞ = e−yxdx = (−1)m f(−m)(x)e−yxdx y dym xm (cid:18) (cid:19) Z0 ! Z0 blog 1 = (−1)mF(y) =(−1)m a +(−1)m y +o 1 , ym+1 ym+1 ym+(cid:16)1(cid:17) ym+1 (cid:18) (cid:19) as y → 0+. Integrating m-times the above asymptotic formula and multi- plying by y we get m a b 1 b 1 T(y) = − + log +o(1) , y → 0+ . m! m! k m! y k=1 (cid:18) (cid:19) X Thus,τ satisfiesthehypothesisofTheorem3.3,and(4.6)followsatonce. (cid:3) We also have, Corollary 4.1. Let f ∈ D′(R) be supported in [0,∞). Suppose that f′ has the distributional asymptotic behavior δ(x) logλ b H(x) 1 f′(λx) = a +b δ(x)+ Pf +o . λ λ λ x λ (cid:18) (cid:19) (cid:18) (cid:19) If (4.3) holds, then f(x)= a+blogx+o(1) (C,m), x → ∞.