The converse of Abel’s theorem on power series. (English) JFM 42.0276.01

There are two reviews of this item, the original review from 1912 by K. Knopp and a “looking back” review from 2013 by Hervé Queffélec.
First review by K. Knopp (Berlin) (1912):
Der Abelsche Grenzwertsatz für Potenzreihen besagt: Wenn \(\sum a_n\) konvergiert, so ist \(\lim\sum a_nx^n=\sum a_n\), wenn \(x\) sich radial (oder auf einem anderen Strahle aus dem Innern des Einheitskreises) dem Punkte \(+1\) nähert. Dieser Satz ist bekanntlich nicht umkehrbar, d. h. aus der Existenz von \(\lim\sum a_n x^n\) bei der erwähnten Annäherung von \(x\) an \(+1\) folgt keineswegs die Konvergenz von \(\sum a_n\) (Beispiel: \(\frac1{1+x}=\sum(-1)^nx^n\)). A. Tauber [Monatsh. f. Math. 8, 273–277 (1897; JFM 28.0221.02)] hat entdeckt, daß, wenn die \(a_n\) noch gewisse Bedingungen erfüllen, doch auf die Konvergenz von \(\sum a_n\) geschlossen werden kann. Die von Tauber angegebene Bedingung ist, daß \(\lim na_n=0\) ist, eine später zu \(\lim\frac1n(a_1+2a_2+\cdots+na_n)=0\) erweiterte Bedingung, welch letztere für die Konvergenz von \(\sum a_n\) sogar notwendig ist. Die Frage war angeregt, aber bisher noch nicht erledigt worden, ob aus der noch weniger verlangenden Voraussetzung, daß \(|na_n|<K\) (d. h. kleiner als eine von \(n\) unabhängige Konstante) bleibt, dasselbe sich folgern läßt.
Die sehr tief gelegene Tatsache, das dies möglich ist, beweist J. E. Littlewood in der vorliegenden Arbeit; er beweist also den Satz: Wenn \(\lim\sum a_nx^n\) in dem oben genannten Sinne vorhanden ist, und wenn \(|na_n|<K\) bleibt, so ist \(\sum a_n\) konvergent.
Darüber hinaus wird gezeigt, daß die Bedingung \(|na_n|<K\) das geringste ist, was zur Erzwingung der Konvergenz gefordert werden muß: ist \(\varphi(n)\) eine (beliebig schwach) ins Unendliche wachsende positive Funktion, so lassen sich Koeffizienten \(a_n\) angeben, für die stets \(|na_n|<\varphi(n)\) bleibt, und für die \(\lim\sum a_nx^n\) existiert, während \(\sum a_n\) divergiert.
Die Beweise werden nicht für Potenzreihen, sondern allgemein für beliebige Dirichletsche Reihen der Form \(\sum a_ne^{-\lambda_nx}\) gegeben.
Zum Schluß werden einige Anwendungen auf Potenzreihen und Fouriersche Reihen gegeben.
Looking-back review by Hervé Queffélec (Lille) (2013):
In 1897, the Austrian mathematician A. Tauber published a short paper (5 pages) entitled “Ein Satz aus der Theorie der unendlichen Reihen” [Monatsh. Math. 8, 273–277 (1897; JFM 28.0221.02)], which can be summarized as follows:
(1) If a series \(\sum a_n\) converges, i.e., if \(S_N:=\sum_{n=0}^N a_n\) converges, then
\[ \lim_{x\to 1^-} \sum_{n=0}^\infty a_n x^n =: \lim_{x\to 1^-}f(x)=\sum_{n=0}^\infty a_n \]
by a theorem of Abel.
(2) If a series \(\sum a_n\) converges, then \[ \lim_{N\to \infty} \frac{1}{N}\sum_{n=0}^{N} n a_n = 0 \]
by a theorem of Kronecker.
(3) Both converses are false; the conclusions of either Abel’s or Kronecker’s theorems may hold, and the series \(\sum a_n\) yet diverge. But if we assume both conclusions, then the series \(\sum a_n\) does converge, and we thus have a necessary and sufficient condition for the convergence of a series of complex numbers.
The paper contains two theorems, labelled A and B. Theorem B is devoted to the special case \( na_n=o(1)\) and then Theorem A is the general case, which is reduced after some effort to Theorem B. The proof of Theorem B contains a trick (\(S_N-f(1-1/N)\to 0\)) but is technically easy, and obviously extends (even if Tauber does not mention it) to the quite often encountered case \(\lim_{N\to \infty} \frac{1}{N}\sum_{n=0}^{N} n | a_n| = 0\). Theorem A is slightly more difficult (typically the kind of proof you should prepare before a course if you do not want to be ridiculous). All in all, the paper is nice, even if it contains no example or application.
In 1911, motivated by a question of G. H. Hardy, who had just proved a similar result for the so-called Cesàro summation process, the English mathematician J. E. Littlewood published a longer paper (15 pages) entitled “The converse of Abel’s theorem on power series”, the paper under review, in which he replaces the second assumption of Theorem B of Tauber by the assumption \(na_n=O(1)\). In his introduction, he quotes Tauber and qualifies his work as “remarkable”, in the way a tennis player having just severely defeated you qualifies your back-hand as remarkable...
Indeed, the proof of the young Littlewood (26 years old at that time) turns out to be incredibly more difficult and elaborate than that of Tauber (and that of Hardy as well!) and somehow heralds the great analyst he will be. His dense paper, which should be read again by today’s mathematicians, doubtlessly with great profit, contains at least three fascinating issues, namely:
1. An analysis of Tauber’s proof, which in fact proves, according to Littlewood, that when \(na_n=o(1)\), the respective cluster sets \(E_f\) and \(E_S\) of \(f(x)\) as \(x\to 1^-\) and of \(S_N\) as \(N\to \infty\) are the same. And the detailed study of a non-trivial example (\(a_n=n^{-1-i\alpha}\) with \(\alpha\) a non-zero real number, observe that \(n| a_n|=1\)), showing how different the situation can be under the conditions \(na_n=o(1)\) or \(na_n=O(1)\). Indeed, in that case, Littlewood shows that \(E_f\) and \(E_S\) are circles with the same center \(\zeta(1+i\alpha)\) and with respective radii \(r_\alpha\) and \(R_\alpha\) such that \(r_\alpha<R_\alpha\).
2. The “tour de force” of the paper: the positive answer to Hardy’s question in Theorem B. New ideas (use of a parameter and of a degree of freedom in an apparently quite rigid problem) appear. Indeed, setting \(x=e^{-\varepsilon}\) and \(S(t)=\sum_{n\leq t}a_n\), Littlewood first reformulates the assumption under the form \[ \lim_{\varepsilon\to 0^+} \varepsilon \int_{0}^\infty e^{-\varepsilon t} S(t)\,dt=:l \] exists. Then, he forces the introduction of a parameter \(r\) by showing that
\[ \lim_{\varepsilon \to 0}\varepsilon^{r+1}\int_{0}^\infty t^r e^{-\varepsilon t} S(t)\,dt=lr! \] for each non-negative integer \(r\). To that effect, the author needs a Theorem A on differentiable functions (essentially the fact that if \(\Phi(x)\to s\in \mathbb{C}\) and \(\Phi'(x), \Phi''(x)\) are bounded near infinity, then \(\Phi'(x)\to 0\)) which is indeed quite simple and was already known (Hadamard, Kneser). In any case, after some delicate estimates on integrals, Littlewood is able to derive that \[ \lim_{T\to \infty} \frac{1}{T}\int_{0}^T S(u)du=l \] and then concludes with the help of the Hardy result already mentioned for Cesàro summation.
3. A proof of the optimality of the condition \(na_n=O(1)\) in Theorem C, under the following form: If \(\varphi_n\to +\infty\), there exists a series \(\sum a_n\) such that
\(\displaystyle | a_n|\leq \frac{\varphi_n}{n},\quad n\geq 1\).
\(\displaystyle \frac{S_0+\cdots +S_N}{N}\to 0, \hbox{\ implying}\;f(x)\to 0\) as \(x\to 1^-\).
\(S_N\) oscillates.

Indeed, Littlewood’s example is (essentially) the following:
\[ \Phi(n)=\sum_{j=1}^n \frac{\varphi_j}{j},\;\; a_n= e^{i\Phi(n)} - e^{i\Phi(n-1)},\;\; S_n=e^{i\Phi(n)} \hbox{\;for}\;n\geq 1. \] The second item is the non-trivial one (its “implication” is the extension by Frobenius of Abel’s theorem), and requires fairly sharp estimates of independent interest on the exponential sums \(S_0+\cdots +S_N=\sum_{n=1}^N e^{i\Phi(n)}\). Littlewood achieves those estimates with a very simple proof, which announces the van der Corput and Kuzmin-Landau estimates in Number Theory, and precedes them by more than 20 years! Indeed, an analysis of the Kuzmin-Landau proof lets it appear as an improvement of Littlewood’s original method.
Let us finish with a few comments:
(1) The example \(a_n=(-1)^n e^{\sqrt n}\) , which is Abel summable, but by no means Cesàro, or higher order Cesàro, summable (due to the severe increase of \(| a_n|\)) can heuristically explain why Littlewood’s result is so much more difficult than Hardy’s one. The assumption of Abel summability is very weak!
(2) Littlewood uses a much more general context than that of Taylor series, that of Dirichlet series \(\sum a_n e^{-\lambda_{n} x}\), in Theorem B. In that general context, one has to assume that \(a_n=O(\frac{\lambda_n-\lambda_{n-1}}{\lambda_n}\)). This generality imposes some restrictions on the exponents \(\lambda_n\), namely \(\frac{\lambda_{n+1}}{\lambda_n}\to 1\), and will be the origin of the so-called problem of “high indices”, later solved by Hardy and Littlewood. Actually \(a_n=O(\frac{\lambda_n-\lambda_{n-1}}{\lambda_n})\) is optimal as well in the general framework of Dirichlet series.
(3) Theorem A of Littlewood can be completely dispensed with, even if heuristically that was the starting point of his discovery. The work of Karamata has shown how to introduce for free (changing \(\varepsilon\) into \(\varepsilon r\)) the degree of freedom which Littlewood has to buy, in writing the assumption under the form
\[ \lim_{\varepsilon\to 0^+} \varepsilon \int_{0}^\infty e^{-\varepsilon r t} S(t)\,dt=l\int_{0}^\infty e^{-rt}\,dt \] for any \(r>0\) and then in linearizing:
\[ \lim_{\varepsilon\to 0^+} \varepsilon \int_{0}^\infty e^{-\varepsilon t} P(e^{-\varepsilon t}) S(t)\,dt=l\int_{0}^\infty e^{-t}P(e^{-t})\,dt \]
for every polynomial \(P\).
(4) This sensational paper of Littlewood immediately attracted the interest of Hardy (34 years old at that time), and gave rise to the famous Hardy-Littlewood collaboration for more than thirty years. One important aspect of this collaboration was to give more credit to Tauber and to put his result into a general context, not only thinking of it as a criterion for convergence of series. As a result, this led to the general theory of “Tauberian theorems”, like the Karamata, Wiener-Ikehara, Newman, Delange, Erdős-Feller-Pollard …theorems, and in particular provided new and simple proofs of the Prime Number Theorem for primes (in an arithmetic progression).
(5) Finally, J. Korevaar published nearly one century later a huge work “Tauberian Theory, a Century of Developments” under the form of a 500 pages long research book [Grundlehren der Mathematischen Wissenschaften 329. Berlin: Springer (2004; Zbl 1056.40002)], which summarizes the discoveries of the Tauberian theory as far as 2004. The posterity of Tauber’s and Littlewood’s early papers has thus proved to be remarkable!


40E05 Tauberian theorems
40-03 History of sequences, series, summability
01A60 History of mathematics in the 20th century
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