Alfred Tauber

Alfred Tauber
Alfred Tauber
	Abelian and tauberian theorems
	Scientific career
Fields	Mathematics
Institutions	TU Wien; University of Vienna
Theses	Über einige Sätze der Gruppentheorie (1889); Über den Zusammenhang des reellen und imaginären Teiles einer Potenzreihe (1891);
Doctoral advisor	Gustav von Escherich; Emil Weyr;

Alfred Tauber (5 November 1866 – 26 July 1942)

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

.

Life and academic career

Born in Pressburg,

Vienna University in 1884, obtained his Ph.D. in 1889,^[3]^[4] 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 the

extraordinary professor. However, he continued lecturing as a privatdozent until 1938,^[3]^[6] when he was forced to resign as a consequence of the "Anschluss".^[7] On 28–29 June 1942, he was deported with transport IV/2, č. 621 to Theresienstadt,^[3]^[5]^[8] where he was murdered on 26 July 1942.^[1]

Work

the theory of quaternions, analytic and descriptive geometry.^[10] Tauber's most important scientific contributions belong to the first of his research areas,^[11] even if his work on potential theory has been overshadowed by the one of Aleksandr Lyapunov.^[3]

Tauberian theorems

His most important article is (

Tauberian to identify this class of theorems.^[15] Describing with a little more detail Tauber's 1897 work, it can be said that his main achievements are the following two theorems:^[16]^[17]

Tauber's first theorem.

Abel summable to sum

s

, i.e.

lim x \to 1 - \sum +\infty n =0 a n x n = s

, and if

a n = ο (n -1)

, then

\sum a k

converges

to

s

.

This theorem is, according to Korevaar (2004, p. 10),^[19] the forerunner of all Tauberian theory: the condition $a n = ο (n -1)$ is the first Tauberian condition, which later had many profound generalizations.^[20] In the remaining part of his paper, by using the theorem above,^[21] Tauber proved the following, more general result:^[22]

Tauber's second theorem.^[23] The series

\sum a n

converges to sum

s

if and only if the two following conditions are satisfied:

$\sum a n$ is Abel summable and
$\sum n k =1 k a k = ο (n)$ .

This result is not a trivial consequence of

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,^[25] though it has its rightful place in all detailed developments of summability of series.^[23]^[25]

Contributions to the theory of Hilbert transform

Frederick W. King (
imaginary part $ψ$ of a power series $f$ ,^[27]^[28]

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

where

$z = re i θ$ with $r = | z |$ being the
complex variable
,

$c k r k = a k + i b k$ for every natural number $k$ ,^[29]

$φ (θ) = \sum +\infty k =1 a k cos(kθ) - b k sin(kθ)$ and $ψ (θ) = \sum +\infty k =1 a k sin(kθ) + b k cos(kθ)$ are trigonometric series and therefore periodic functions, expressing the real and imaginary part of the given power series.

Under the
convergence radius
$R f$ of the power series $f$ , Tauber proves that $φ$ and $ψ$ satisfy the two following equations:

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

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

Assuming then $r = R f$ , he is also able to prove that the above equations still hold if $φ$ and $ψ$ are only
absolutely integrable:^[30] 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:^[31]

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

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
open disk
if and only if the two following conditions are satisfied

the function $[φ (θ - α) - φ (θ + α)]/α$ is
neighborhood
of the point $α = 0$ , and

the function $ψ (θ)$ satisfies (2).

Selected publications

Tauber, Alfred (1891), "Über den Zusammenhang des reellen und imaginären Theiles einer Potenzreihe" [On the relation between real and imaginary part of a power series], S2CID 120241651
.

Tauber, Alfred (1895), "Ueber die Werte einer analytischen Function längs einer Kreislinie" [On the values of an analytic function along a circular perimeter],
Jahresbericht der Deutschen Mathematiker-Vereinigung, 4: 115, archived from the original
on 2015-07-01, retrieved 2014-07-16.

Tauber, Alfred (1897), "Ein Satz aus der Theorie der unendlichen Reihen" [A theorem about infinite series], S2CID 120692627
.

Tauber, Alfred (1898), "Über einige Sätze der Potentialtheorie" [Some theorems of potential theory], S2CID 124400762
.

Tauber, Alfred (1920), "Über konvergente und asymptotische Darstellung des Integrallogarithmus" [On convergent and asymptotic representation of the logarithmic integral function], S2CID 119967249
.

Tauber, Alfred (1922), "Über die Umwandlung von Potenzreihen in Kettenbrüche" [On the conversion of power series into continued fractions], S2CID 122501264
.

See also

Actuarial science

Hardy–Littlewood tauberian theorem

Summability theory

Notes

^ ^a ^b ^c The death date is reported in (Sigmund 2004, p. 33) and also in Tauber's VIAF record Archived 2018-09-18 at the Wayback Machine, 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.

summability
" area.

^ ^a ^b ^c ^d ^e ^f ^g (Hlawka 2007).

^ According to Hlawka (2007), he wrote his doctoral dissertation in 1888.

^ ^a ^b ^c (Pinl & Dick 1974, pp. 202–203).

actuarial mathematics
by his low pension.

^ (Sigmund 2004, p. 21 and p. 28).

^ (Fischer et al. 1990, p. 812, footnote 14).

^ See the results of Jahrbuch query: "au = (TAUBER, A*)".

^ In the exact authors' words, "Unendliche Reihen, Fouriersche Reihen, Kugelfunktionen, Quaternionen,..., Analitische und Darstellende Geometrie" (Pinl & Dick 1974, p. 202).

^ According to Hlawka's classification (2007).

^ 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).

^ See for example (Hardy 1949, p. 149) and (Korevaar 2004, p. 6).

^ See (Hardy 1949, p. 149), (Hlawka 2007) and (Lune 1986, p. 2 §1.1 "Tauber's first theorem").

^ 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).

^ 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").

^ The Landau little– $ο$ notation is used in the following description.

^ See for example (Hardy 1949, p. 149), (Korevaar 2004, p. 10) and (Lune 1986, p. 2, §1.1 "Tauber's first theorem").

^ 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).

^ See (Hardy 1949, p. 149), Chatterji (1984, p. 169 and p. 172) and (Korevaar 2004, p. 6).

^ 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.

^ (Chatterji 1984, p. 169 theorem A), (Korevaar 2004, p. 11).

^ ^a ^b See for example (Hardy 1949, p. 150), (Korevaar 2004, p. 11) and (Lune 1986, p. 4, §1.2 "Tauber's second theorem").

^ 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).

^ ^a ^b Again according to Chatterji (1984, p. 172).

^ In King's words (2009, p.3), "In hindsight, perhaps the transform should bear the names of the three aforementioned authors".

^ The analysis presented closely follows (King 2009, p. 131), which in turn follows (Tauber 1891, pp. 79–80).

^ See also the short research announce (Tauber 1895).

^ As King (2009, p. 131) notes, this non-standard definition of the real and imaginary part of the $k$ th complex coefficient of a power series is purposefully introduced in order to hidden ("suppress") the functional dependence of $φ$ and $ψ$ on $r$ .

^ This means that $φ, ψ$ $\in$ $L 1$ .

^ (King 2009, p. 131).

References

Biographical and general references

Zbl 0544.01021

Fischer, Gerd;
Zbl 0706.01002
.

Pinl, Maximilian;
Zbl 0281.01013
.

Complete Dictionary of Scientific Biography, New York: Charles Scribner's Sons
, retrieved 27 February 2016.

Sigmund, Karl (2004), "Failing Phoenix: Tauber, Helly, and Viennese life insurance", Zbl 0849.01036
.

Scientific references

Chatterji, S. D. (1984), "Tauber's theorem – a few historical remarks", in Chatterji, S. D. (ed.), Jahrbuch Überblicke Mathematik, Mathematical surveys, vol. 17,
Zbl 0556.01005
.

ISBN 0828403341
.

JFM 44.0283.01
.

King, Frederick W. (2009), Hilbert transforms. Volume 1, Encyclopedia of Mathematics and its Applications, vol. 124,
Zbl 1188.44005
.

Korevaar, Jacob (2004), Tauberian theory. A century of developments, Grundlehren der Mathematischen Wissenschaften, vol. 329,
Zbl 1056.40002
.

Lune, J. van de (1986), An introduction to Tauberian theory: from Tauber to Wiener, CWI Syllabus, vol. 12,
Zbl 0636.40002
.

External links

O'Connor, John J.; Robertson, Edmund F., "Alfred Tauber", MacTutor History of Mathematics Archive, University of St Andrews

Alfred Tauber at encyclopedia.com

Alfred Tauber at the Mathematics Genealogy Project

Authority control databases
International

ISNI

VIAF

National

Germany

Czech Republic

Academics

MathSciNet

Mathematics Genealogy Project

zbMATH

People

Deutsche Biographie

Retrieved from "https://en.wikipedia.org/w/index.php?title=Alfred_Tauber&oldid=1167236960"

[death.date-1] The death date is reported in (Sigmund 2004, p. 33) and also in Tauber's VIAF record Archived 2018-09-18 at the Wayback Machine, 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] summability
" area.

[Hlawka.2007-3] ^ ^a ^b ^c ^d ^e ^f ^g (Hlawka 2007).

[4] According to Hlawka (2007), he wrote his doctoral dissertation in 1888.

[Pinl.Dick.1974-5] (Pinl & Dick 1974, pp. 202–203).

[6] tuarial mathematics
by his low pension.

[7] (Sigmund 2004, p. 21 and p. 28).

[Fischer.Hirzebruch.Scharlau.1990-8] (Fischer et al. 1990, p. 812, footnote 14).

[9] See the results of Jahrbuch query: "au = (TAUBER, A*)".

[10] In the exact authors' words, "Unendliche Reihen, Fouriersche Reihen, Kugelfunktionen, Quaternionen,..., Analitische und Darstellende Geometrie" (Pinl & Dick 1974, p. 202).

[11] According to Hlawka's classification (2007).

[12] 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).

[13] See for example (Hardy 1949, p. 149) and (Korevaar 2004, p. 6).

[14] See (Hardy 1949, p. 149), (Hlawka 2007) and (Lune 1986, p. 2 §1.1 "Tauber's first theorem").

[15] 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).

[16] 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").

[17] The Landau little– $ο$ notation is used in the following description.

[18] See for example (Hardy 1949, p. 149), (Korevaar 2004, p. 10) and (Lune 1986, p. 2, §1.1 "Tauber's first theorem").

[19] 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).

[20] See (Hardy 1949, p. 149), Chatterji (1984, p. 169 and p. 172) and (Korevaar 2004, p. 6).

[21] 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.

[22] (Chatterji 1984, p. 169 theorem A), (Korevaar 2004, p. 11).

[2nd.theorem-23] See for example (Hardy 1949, p. 150), (Korevaar 2004, p. 11) and (Lune 1986, p. 4, §1.2 "Tauber's second theorem").

[24] 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).

[Chatterji.1984.172-25] Again according to Chatterji (1984, p. 172).

[26] In King's words (2009, p.3), "In hindsight, perhaps the transform should bear the names of the three aforementioned authors".

[27] The analysis presented closely follows (King 2009, p. 131), which in turn follows (Tauber 1891, pp. 79–80).

[28] See also the short research announce (Tauber 1895).

[29] As King (2009, p. 131) notes, this non-standard definition of the real and imaginary part of the $k$ th complex coefficient of a power series is purposefully introduced in order to hidden ("suppress") the functional dependence of $φ$ and $ψ$ on $r$ .

[30] This means that $φ, ψ$ $\in$ $L 1$ .

[31] (King 2009, p. 131).

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