# Alfred Tauber

Not to be confused with the medicine and philosophy professor Alfred I. Tauber (born 1947).
Alfred Tauber
Born November 5, 1866
Pressburg, Austrian Empire (today Bratislava, Slovakia)
Died July 26, 1942 (aged 75)[1]
Nationality Austrian
Fields Mathematics
Institutions TH Vienna
University of Vienna
Alma mater University of Vienna
Theses
• Über einige Sätze der Gruppentheorie (1889)
• Über den Zusammenhang des reellen und imaginären Teiles einer Potenzreihe (1891)
Known for Abelian and tauberian theorems

Alfred Tauber (November 5, 1866 – July 26, 1942)[1] was an Austrian and Slovak mathematician, known for his contribution to mathematical analysis and to the theory of functions of a complex variable: he is the eponym of an important class of theorems with applications ranging from mathematical and harmonic analysis to number theory.[2] He was murdered in the Theresienstadt concentration camp.

Born in Bratislava,[3] he began studying mathematics at Vienna University in 1884, obtained his Ph.D. in 1889,[4][5] and his habilitation in 1891. Starting from 1892, he worked as chief mathematician at the Phönix insurance company until 1908, when he became an a.o. professor at Vienna University, though, already from 1901, he had been honorary professor at TH Vienna and director of its insurance mathematics chair.[6] In 1933 he was awarded the Grand Decoration of Honour in Silver for Services to the Republic of Austria,[6] and retired as emeritus extraordinary professor. However, He continued lecturing as a privatdozent until 1938,[4][7] when he was forced to resign as a consequence of the "Anschluss".[8] On 28–29 June 1942 he was deported with transport IV/2, č. 621 to Theresienstadt,[4][6][9] where he was murdered on 26 July 1942.[1]

## Work

Pinl & Dick (1974, p. 202) list 35 publications in the bibliography appended to his obituary, and also a search performed on the "Jahrbuch über die Fortschritte der Mathematik" database results in a list 35 mathematical works authored by him, spanning a period of time from 1891 to 1940.[10] However, Hlawka (2007) cites two papers on actuarial mathematics which do not appear in these two bibliographical lists and Binder's bibliography of Tauber's works (1984, pp. 163–166), while listing 71 entries including the ones in the bibliography of Pinl & Dick (1974, p. 202) and the two cited by Hlawka, does not includes the short note (Tauber 1895) so the exact number of his works is not known. According to Hlawka (2007), his scientific research can be divided into three areas: the first one comprises his work on the theory of functions of a complex variable and on potential theory, the second one includes works on linear differential equations and on the Gamma function, while the last one includes his contributions to actuarial science.[4] Pinl & Dick (1974, p. 202) give a more detailed list of research topics Tauber worked on, though it is restricted to mathematical analysis and geometric topics: some of them are infinite series, Fourier series, spherical harmonics, the theory of quaternions, analytic and descriptive geometry.[11] Tauber's most important scientific contributions belong to the first of his research areas,[12] even if his work on potential theory has been overshadowed by the one of Aleksandr Lyapunov.[4]

### Tauberian theorems

His most important article is (Tauber 1897).[4] In this paper, he succeeded in proving a converse to Abel's theorem for the first time:[13] this result was the starting point of numerous investigations,[4] leading to the proof and to applications of several theorems of such kind for various summability methods. The statement of these theorems has a standard structure: if a series ∑ an is summable according to a given summability method and satisfies an additional condition, called "Tauberian condition",[14] then it is a convergent series.[15] Starting from 1913 onward, G. H. Hardy and J. E. Littlewood used the term Tauberian to identify this class of theorems.[16] Describing with a little more detail Tauber's 1897 work, it can be said that his main achievements are the following two theorems:[17][18]

Tauber's first theorem.[19] If the series ∑ an is Abel summable to sum s, i.e. limx→ 1  +∞
n=0

an x n =  s
, and if an = ο(n−1), then ∑ ak converges to s.

This theorem is, according to Korevaar (2004, p. 10),[20] the forerunner of all Tauberian theory: the condition an = ο(n−1) is the first Tauberian condition, which later had many profound generalizations.[21] In the remaining part of his paper, by using the theorem above,[22] Tauber proved the following, more general result:[23]

Tauber's second theorem.[24] The series ∑ an converges to sum s if and only if the two following conditions are satisfied:
1. ∑ an is Abel summable and
2. n
k=1

k ak = ο(n)
.

This result is not a trivial consequence of Tauber's first theorem.[25] The greater generality of this result respect to the former one is due to the fact it proves the exact equivalence between ordinary convergence on one side and Abel summability (condition 1) jointly with Tauberian condition (condition 2) on the other. Chatterji (1984, pp. 169–170) claims that this latter result must have appeared to Tauber much more complete and satisfying respect to the former one as it states a necessary and sufficient condition for the convergence of a series while the former one was simply a stepping stone to it: the only reason why Tauber's second theorem is not mentioned very often seems to be that it has no profound generalization as the first one has,[26] though it has its rightful place in all detailed developments of summability of series.[24][26]

### Contributions to the theory of Hilbert transform

Frederick W. King (2009, p. 3) writes that Tauber contributed at an early stage to theory of the now called "Hilbert transform", anticipating with his contribution the works of Hilbert and Hardy in such a way that the transform should perhaps bear their three names.[27] Precisely, Tauber (1891) considers the real part φ and imaginary part ψ of a power series f,[28][29]

${\displaystyle f(z)=\sum _{k=1}^{+\infty }c_{k}z^{k}=\varphi (\theta )+\mathrm {i} \psi (\theta )}$

where

Under the hypothesis that r is less than the convergence radius Rf of the power series f, Tauber proves that φ and ψ satisfy the two following equations:

(1)     ${\displaystyle \varphi (\theta )={\frac {1}{2\pi }}\int _{0}^{\pi }\left\{\psi (\theta +\phi )-\psi (\theta -\phi )\right\}\cot \left({\frac {\phi }{2}}\right)\,\mathrm {d} \phi }$
(2)     ${\displaystyle \psi (\theta )=-{\frac {1}{2\pi }}\int _{0}^{\pi }\left\{\varphi (\theta +\phi )-\varphi (\theta -\phi )\right\}\cot \left({\frac {\phi }{2}}\right)\mathrm {d} \phi }$

Assuming then r = Rf, he is also able to prove that the above equations still hold if φ and ψ are only absolutely integrable:[31] this result is equivalent to defining the Hilbert transform on the circle since, after some calculations exploiting the periodicity of the functions involved, it can be proved that (1) and (2) are equivalent to the following pair of Hilbert transforms:[32]

${\displaystyle \varphi (\theta )={\frac {1}{2\pi }}\int _{-\pi }^{\pi }\psi (\phi )\cot \left({\frac {\theta -\phi }{2}}\right)\mathrm {d} \phi \qquad \psi (\theta )={\frac {1}{2\pi }}\int _{-\pi }^{\pi }\varphi (\phi )\cot \left({\frac {\theta -\phi }{2}}\right)\mathrm {d} \phi }$

Finally, it is perhaps worth pointing out an application of the results of (Tauber 1891), given (without proof) by Tauber himself in the short research announce (Tauber 1895):

the complex valued continuous function φ(θ) + iψ(θ) defined on a given circle is the boundary value of a holomorphic function defined in its open disk if and only if the two following conditions are satisfied
1. the function [φ(θ − α) − φ(θ + α)]/α is uniformly integrable in every neighborhood of the point α = 0, and
2. the function ψ(θ) satisfies (2).

## Notes

1. ^ a b c The death date is reported in (Sigmund 2004, p. 33) and also in Tauber's VIAF record, line 678: Sigmund (2004, pp. 31–33) also gives a description of the events of the last years of Tauber's life, up to the days of his deportation.
2. ^ The 2010 Mathematics Subject Classification has two entries on Tauberian theorems: the entry 11M45, belonging to the "Number theory" area, and the entry 40E05, belonging to the "Sequences, series, summability" area.
3. ^ At the time when it was still in the Kingdom of Hungary and called "Pozsony".
4. (Hlawka 2007).
5. ^ According to Hlawka (2007), he wrote his doctoral dissertation in 1888.
6. ^ a b c (Pinl & Dick 1974, pp. 202–203).
7. ^ Sigmund (2004, p. 2) states that he was forced to keep holding his course on actuarial mathematics by his low pension.
8. ^ (Sigmund 2004, p. 21 and p. 28).
9. ^ (Fischer et al. 1990, p. 812, footnote 14).
10. ^ See the results of Jahrbuch query: "au = (TAUBER, A*)".
11. ^ In the exact authors' words, "Unendliche Reihen, Fouriersche Reihen, Kugelfunktionen, Quaternionen,..., Analitische und Darstellende Geometrie" (Pinl & Dick 1974, p. 202).
12. ^ According to Hlawka's classification (2007).
13. ^ See for example (Hardy 1949, p. 149), (Hlawka 2007), (Korevaar 2004, p. VII, p. 2 and p. 10), (Lune 1986, p. 2, §1.1 "Tauber's first theorem") and (Sigmund 2004, p. 21).
14. ^ See for example (Hardy 1949, p. 149) and (Korevaar 2004, p. 6).
15. ^ See (Hardy 1949, p. 149), (Hlawka 2007) and (Lune 1986, p. 2 §1.1 "Tauber's first theorem").
16. ^ See (Korevaar 2004, p. 2) and (Sigmund 2004, p. 21): Korevaar precises that the locution "Tauberian theorems" was first used in the short note (Hardy & Littlewood 1913).
17. ^ See (Hardy 1949, p. 149 and p. 150), (Korevaar 2004, p. 10 and p. 11) and (Lune 1986, p. 2, §1.1 "Tauber's first theorem" and p. 4, §1.1 "Tauber's second theorem").
18. ^ The Landau little–ο notation is used in the following description.
19. ^ See for example (Hardy 1949, p. 149), (Korevaar 2004, p. 10) and (Lune 1986, p. 2, §1.1 "Tauber's first theorem").
20. ^ See also (Lune 1986, p. 2, §1.1 "Tauber's first theorem") and (Hardy 1949, p. 149): Sigmund (2004, p. 21) incorrectly attributes this role to Tauber's second theorem. See also the analysis by Chatterji (1984, pp. 169–170 and p. 172).
21. ^ See (Hardy 1949, p. 149), Chatterji (1984, p. 169 and p. 172) and (Korevaar 2004, p. 6).
22. ^ See (Chatterji 1984, p. 169 theorem B), (Lune 1986, p. 4, §1.2 "Tauber's second theorem") and the remark by Korevaar (2004, p. 11): Hardy (1949, pp. 150–152) proves this theorem by proving a more general one involving Riemann–Stieltjes integrals.
23. ^ (Chatterji 1984, p. 169 theorem A), (Korevaar 2004, p. 11).
24. ^ a b See for example (Hardy 1949, p. 150), (Korevaar 2004, p. 11) and (Lune 1986, p. 4, §1.2 "Tauber's second theorem").
25. ^ According to Chatterji (1984, p. 172): see also the proofs of the two theorems given by Lune (1986, chapter 1, §§1.1–1.2, pp. 2–7).
26. ^ a b Again according to Chatterji (1984, p. 172).
27. ^ In King's words (2009, p.3), "In hindsight, perhaps the transform should bear the names of the three aforementioned authors".
28. ^ The analysis presented closely follows (King 2009, p. 131), which in turn follows (Tauber 1891, pp. 79–80).