View source for Alfred Tauber

{{Short description|Hungarian mathematician (1866–1942)}}
{{distinguish|text=the medicine and philosophy professor [[Alfred I. Tauber]] (born 1947)}}
{{Infobox scientist
 | name              = Alfred Tauber
 | image             = Alfred Tauber.jpg
 | image_size        =
 | caption           =
 | birth_date        = {{birth date|1866|11|5|df=y}}
 | birth_place       = Pressburg, [[Kingdom of Hungary (1526–1867)|Kingdom of Hungary]], [[Austrian Empire]] (today [[Bratislava]], [[Slovakia]])
 | death_date        = {{death date and age|1942|07|26|1866|11|05|df=y}}<ref name="death.date">The death date is reported in {{harv|Sigmund|2004|p=33}} and also in [http://viaf.org/processed/NKC|jo2007416025 Tauber's VIAF record] {{Webarchive|url=https://web.archive.org/web/20180918151340/http://viaf.org/processed/NKC%7Cjo2007416025 |date=2018-09-18 }}, line 678: {{harvtxt|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.</ref>
 | death_place       = [[Theresienstadt Ghetto|Theresienstadt concentration camp]], [[Czechoslovakia]]
 | nationality       = Austrian
 | fields            = [[Mathematics]]
 | workplaces        = [[TU Wien]]<br>[[University of Vienna]]
 | alma_mater        = [[University of Vienna]]
 | thesis1_title     = Über einige Sätze der Gruppentheorie
 | thesis1_url       =
 | thesis1_year      = 1889
 | thesis2_title     = Über den Zusammenhang des reellen und imaginären Teiles einer Potenzreihe
 | thesis2_url       =
 | thesis2_year      = 1891
 | doctoral_advisor  = {{Plainlist|
* [[Gustav von Escherich]]
* [[Emil Weyr]] }}
 | doctoral_students =
 | known_for         = [[Abelian and tauberian theorems]]
 | awards            =
}}

'''Alfred Tauber''' (5 November 1866 – 26 July 1942)<ref name="death.date"/> was a 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 analysis|mathematical]] and [[harmonic analysis]] to [[number theory]].<ref>The 2010 [[Mathematics Subject Classification]] [https://www.ams.org/mathscinet/msc/msc2010.html?t=&s=Tauberian+theorems&btn=Search&ls=Cs has two entries] on Tauberian theorems: the entry 11M45, belonging to the "Number theory" area, and the entry 40E05, belonging to the "[[Sequence (mathematics)|Sequences]], [[Series (mathematics)|series]], [[summability]]" area.</ref> He was [[History of the Jews in Hungary#Austria-Hungary (1867–1918) |born in Austria-Hungary]], [[History of the Jews in Austria#Modern period |lived in Vienna]], [[First Austrian Republic |Austria]] after the dissolution of the empire, and was deported and murdered for being Jewish when the [[Theresienstadt Ghetto| Theresienstadt concentration camp]] was emptied of Jews in 1942.<ref>[http://www.geometry.net/detail/scientists/tauber_alfred.html Alfred Tauber] at geometry.net, accessed 18 Dec 2024.</ref>

==Life and academic career==
Born in Pressburg, [[Kingdom of Hungary (1526–1867)|Kingdom of Hungary]], [[Austrian Empire]] (now [[Bratislava]], [[Slovakia]]), he began studying mathematics at [[Vienna University]] in 1884, obtained his Ph.D. in 1889,<ref name="Hlawka.2007">{{harv|Hlawka|2007}}.</ref><ref>According to {{harvtxt|Hlawka|2007}}, he wrote his doctoral dissertation in 1888.</ref> 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 [[University of Vienna]], though, already from 1901, he had been honorary professor at [[TU Wien|TU Vienna]] and director of its insurance mathematics chair.<ref name="Pinl.Dick.1974">{{harv|Pinl|Dick|1974|pp=202–203}}.</ref> In 1933, he was awarded the [[Grand Decoration of Honour in Silver for Services to the Republic of Austria]],<ref name="Pinl.Dick.1974"/> and retired as [[emeritus professor|emeritus]] [[extraordinary professor]]. However, he continued lecturing as a [[privatdozent]] until 1938,<ref name="Hlawka.2007"/><ref>{{harvtxt|Sigmund|2004|p=2}} states that he was forced to keep holding his course on [[actuarial mathematics]] by his low pension.</ref> when he was forced to resign as a consequence of the "[[Anschluss]]".<ref>{{harv|Sigmund|2004|loc=p. 21 and p. 28}}.</ref> On 28–29 June 1942, he was deported with transport IV/2, č. 621 to [[Theresienstadt Ghetto|Theresienstadt]],<ref name="Hlawka.2007"/><ref name="Pinl.Dick.1974"/><ref name="Fischer.Hirzebruch.Scharlau.1990">{{harv|Fischer|Hirzebruch|Scharlau|Törnig|1990|loc=p. 812, footnote 14}}.</ref> where he was murdered on 26 July 1942.<ref name="death.date"/>

==Work==
{{harvtxt|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.<ref>See the results of Jahrbuch query: "[http://www.emis.de/cgi-bin/jfmen/MATH/JFM/quick.html?first=1&maxdocs=50&au=Tauber,+A&type=html&format=short au = (TAUBER, A*)]".</ref> However, {{harvtxt|Hlawka|2007}} cites two papers on actuarial mathematics which do not appear in these two bibliographical lists and [[#{{harvid|Binder|1984}}|Binder's bibliography of Tauber's works (1984]], pp.&nbsp;163–166), while listing 71 entries including the ones in the bibliography of {{harvtxt|Pinl|Dick|1974|p=202}} and the two cited by Hlawka, does not includes the short note {{harv|Tauber|1895}} so the exact number of his works is not known. According to {{harvtxt|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 equation]]s and on the [[Gamma function]], while the last one includes his contributions to actuarial science.<ref name="Hlawka.2007"/> {{harvtxt|Pinl|Dick|1974|p=202}} give a more detailed list of research topics Tauber worked on, though it is restricted to [[mathematical analysis]] and [[Geometry|geometric]] topics: some of them are [[infinite series]], [[Fourier series]], [[spherical harmonics]], [[Quaternions|the theory of quaternions]], [[Analytic geometry|analytic]] and [[descriptive geometry]].<ref>In the exact authors' words, "Unendliche Reihen, Fouriersche Reihen, Kugelfunktionen, Quaternionen,..., Analitische und Darstellende Geometrie" {{harv|Pinl|Dick|1974|p=202}}.</ref> Tauber's most important scientific contributions belong to the first of his research areas,<ref>According to [[#{{harvid|Hlawka|2007}}|Hlawka's classification (2007]]).</ref> even if his work on potential theory has been overshadowed by the one of [[Aleksandr Lyapunov]].<ref name="Hlawka.2007" />

===Tauberian theorems===
His most important article is {{harv|Tauber|1897}}.<ref name="Hlawka.2007"/> In this paper, he succeeded in proving a converse to [[Abel's theorem]] for the first time:<ref>See for example {{harv|Hardy|1949|p=149}}, {{harv|Hlawka|2007}}, {{harv|Korevaar|2004|loc= p. VII, p. 2 and p. 10}}, {{harv|Lune|1986|loc=p. 2, §1.1 "Tauber's first theorem"}} and {{harv|Sigmund|2004|p=21}}.</ref> this result was the starting point of numerous investigations,<ref name="Hlawka.2007" /> leading to the proof and to applications of several theorems of such kind for various [[summability method]]s. The statement of these theorems has a standard structure: if a series {{math|&sum;&nbsp;''a{{sub|n}}''}} is summable according to a given summability method and satisfies an additional condition, called "''Tauberian condition''",<ref>See for example {{harv|Hardy|1949|p=149}} and {{harv|Korevaar|2004|p=6}}.</ref> then it is a [[convergent series]].<ref>See {{harv|Hardy|1949|p=149}}, {{harv|Hlawka|2007}} and {{harv|Lune|1986|loc=p. 2 §1.1 "Tauber's first theorem"}}.</ref> Starting from 1913 onward, [[G. H. Hardy]] and [[J. E. Littlewood]] used the term [[Abelian and tauberian theorems|''Tauberian'']] to identify this class of theorems.<ref>See {{harv|Korevaar|2004|p=2}} and {{harv|Sigmund|2004|p=21}}: Korevaar precises that the locution "Tauberian theorems" was first used in the short note {{harv|Hardy|Littlewood|1913}}.</ref> Describing with a little more detail [[#{{harvid|Tauber|2007}}|Tauber's 1897 work]], it can be said that his main achievements are the following two theorems:<ref>See {{harv|Hardy|1949|loc=p. 149 and p. 150}}, {{harv|Korevaar|2004|loc= p. 10 and p. 11}} and {{harv|Lune|1986|loc=p. 2, §1.1 "Tauber's first theorem" and p. 4, §1.1 "Tauber's second theorem"}}.</ref><ref>The [[Big O notation#Little-o notation|Landau little–{{mvar|ο}} notation]] is used in the following description.</ref>

:{{EquationRef|1|Tauber's first theorem}}.<ref>See for example {{harv|Hardy|1949|p=149}}, {{harv|Korevaar|2004|p=10}} and {{harv|Lune|1986|loc=p. 2, §1.1 "Tauber's first theorem"}}.</ref> If the series {{math|&sum;&nbsp;''a{{sub|n}}''}} is [[Abel summability|Abel summable]] to sum {{mvar|s}}, i.e. {{math|lim{{sub|''x''→ 1{{sup|−}}}} &nbsp;{{SubSup|&sum;|s=0|''n''{{=}}0|+∞}}''a{{sub|n}} x{{sup|&nbsp;n}}&nbsp;{{=}}&nbsp; s''}}, and if {{math|''a{{sub|n}}''&nbsp;{{=}}&nbsp;''ο''(''n''{{sup|−1}})}}, then {{math|&sum;&nbsp;''a{{sub|k}}''}} [[Convergent series|converges]] to {{mvar|s}}.

This theorem is, according to {{harvtxt|Korevaar|2004|p=10}},<ref>See also {{harv|Lune|1986|loc=p. 2, §1.1 "Tauber's first theorem"}} and {{harv|Hardy|1949|p=149}}: {{harvtxt|Sigmund|2004|p=21}} incorrectly attributes this role to {{EquationNote|2|Tauber's second theorem}}. See also the analysis by {{harvtxt|Chatterji|1984|loc=pp. 169–170 and p. 172}}.</ref> the forerunner of all Tauberian theory: the condition {{math|''a{{sub|n}}''&nbsp;{{=}}&nbsp;''ο''(''n''{{sup|−1}})}} is the first Tauberian condition, which later had many profound generalizations.<ref>See {{harv|Hardy|1949|p=149}}, {{harvtxt|Chatterji|1984|loc=p. 169 and p. 172}} and {{harv|Korevaar|2004|p=6}}.</ref> In the remaining part of his paper, by using the theorem above,<ref>See {{harv|Chatterji|1984|loc=p. 169 theorem B}}, {{harv|Lune|1986|loc=p. 4, §1.2 "Tauber's second theorem"}} and the remark by {{harvtxt|Korevaar|2004|p=11}}: {{harvtxt|Hardy|1949|pp=150–152}} proves this theorem by proving a more general one involving [[Riemann–Stieltjes integral]]s.</ref> Tauber proved the following, more general result:<ref>{{harv|Chatterji|1984|loc=p. 169 theorem A}}, {{harv|Korevaar|2004|p=11}}.</ref>

:{{EquationRef|2|Tauber's second theorem}}.<ref name="2nd.theorem">See for example {{harv|Hardy|1949|p=150}}, {{harv|Korevaar|2004|p=11}} and {{harv|Lune|1986|loc=p. 4, §1.2 "Tauber's second theorem"}}.</ref> The series {{math|&sum;&nbsp;''a{{sub|n}}''}} converges to sum {{mvar|s}} if and only if the two following conditions are satisfied:
# {{math|&sum;&nbsp;''a{{sub|n}}''}} is Abel summable and
# {{math|{{SubSup|&sum;|s=0|''k''{{=}}1|''n''}}''k a{{sub|k}}''&nbsp;{{=}}&nbsp;''ο''(''n'')}}.

This result is not a trivial consequence of {{EquationNote|1|Tauber's first theorem}}.<ref>According to {{harvtxt|Chatterji|1984|p=172}}: see also the proofs of the two theorems given by {{harvtxt|Lune|1986|loc=chapter 1, §§1.1–1.2, pp. 2–7}}.</ref> The greater generality of this result with 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. {{harvtxt|Chatterji|1984|pp=169–170}} claims that this latter result must have appeared to Tauber much more complete and satisfying respect to the {{EquationNote|1|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,<ref name="Chatterji.1984.172">Again according to {{harvtxt|Chatterji|1984|p=172}}.</ref> though it has its rightful place in all detailed developments of summability of series.<ref name="2nd.theorem" /><ref name="Chatterji.1984.172" />

===Contributions to the theory of Hilbert transform===
{{harvs|txt|first=Frederick W.|last=King|year=2009|loc=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 [[David Hilbert|Hilbert]] and [[Godfrey Harold Hardy|Hardy]] in such a way that the transform should perhaps bear their three names.<ref>In [[#{{harvid|King|2009}}|King's words (2009]], p.3), "''In hindsight, perhaps the transform should bear the names of the three aforementioned authors''".</ref> Precisely, {{harvtxt|Tauber|1891}} considers the [[real part]] {{math|''φ''}} and [[imaginary part]] {{math|''ψ''}} of a [[power series]] {{math|''f''}},<ref>The analysis presented closely follows {{harv|King|2009|p=131}}, which in turn follows {{harv|Tauber|1891|pp=79–80}}.</ref><ref>See also the short research announce {{harv|Tauber|1895}}.</ref>

:<math>f(z)=\sum_{k=1}^{+\infty} c_kz^k =\varphi(\theta)+\mathrm{i} \psi(\theta)</math>

where
*{{math|''z&nbsp;{{=}} re''{{sup|&nbsp;i''θ''}} }} with {{math|''r''&nbsp;{{=}}&nbsp;{{abs|&nbsp;''z''&nbsp;}}}} being the [[absolute value]] of the given [[complex variable]],
*{{math|''c{{sub|k}} r{{sup|&nbsp;k}}''&nbsp;{{=}}&nbsp;''a{{sub|k}}''&nbsp;+&nbsp;i''b{{sub|k}}''}} for every [[natural number]] {{math|''k''}},<ref>As {{harvtxt|King|2009|p=131}} notes, this non-standard definition of the real and imaginary part of the {{math|''k''}}th complex coefficient of a power series is purposefully introduced in order to hidden ("suppress") the functional dependence of {{math|''φ''}} and {{math|''ψ''}} on {{math|''r''}}.</ref>
*{{math|''φ''(''θ'')&nbsp;{{=}}&nbsp;{{SubSup|&sum;|s=0|''k''{{=}}1|+∞}}''a{{sub|k}}''cos(''kθ'')&nbsp;−&nbsp;''b{{sub|k}}''sin(''kθ'')}} and {{math|''ψ''(''θ'')&nbsp;{{=}}&nbsp;{{SubSup|&sum;|s=0|''k''{{=}}1|+∞}}''a{{sub|k}}''sin(''kθ'')&nbsp;+&nbsp;''b{{sub|k}}''cos(''kθ'')}} are [[trigonometric series]] and therefore [[periodic function]]s, expressing the real and imaginary part of the given power series.

Under the [[hypothesis]] that {{math|''r''}} is less than the [[convergence radius]] {{math|''R{{Sub|f}}''}} of the power series {{math|''f''}}, Tauber proves that {{math|''φ''}} and {{math|''ψ''}} satisfy the two following equations:

:{{EquationRef|3|(1){{spaces|5}}}}<math>\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</math>

:{{EquationRef|4|(2){{spaces|5}}}}<math>\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</math>

Assuming then {{math|''r&nbsp;{{=}}&nbsp;R{{Sub|f}}''}}, he is also able to prove that the above equations still hold if {{math|''φ''}} and {{math|''ψ''}} are only [[absolutely integrable]]:<ref>This means that {{math|''φ,&nbsp;ψ''}}{{math|&nbsp;∈&nbsp;}}{{math|[[Lp space|''L''{{Sup|1}}]]}}.</ref> this result is equivalent to defining the [[Hilbert transform#Hilbert transform on the circle|Hilbert transform on the circle]] since, after some calculations exploiting the periodicity of the functions involved, it can be proved that {{EquationNote|3|(1)}} and {{EquationNote|4|(2)}} are equivalent to the following pair of Hilbert transforms:<ref>{{harv|King|2009|p=131}}.</ref>
::<math>
\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
</math>
Finally, it is perhaps worth pointing out an application of the results of {{harv|Tauber|1891}}, given (without proof) by Tauber himself in the short research announce {{harv|Tauber|1895}}: 
:the complex valued [[continuous function]] {{math|''φ''(''θ'')&nbsp;+&nbsp;i''ψ''(''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
# the function {{math|[''φ''(''θ&nbsp;−&nbsp;α'')&nbsp;−&nbsp;''φ''(''θ&nbsp;+&nbsp;α'')]/α}} is [[uniform integrability|uniformly integrable]] in every [[Neighborhood (topology)|neighborhood]] of the point {{math|''α''&nbsp;{{=}}&nbsp;0}}, and
# the function {{math|''ψ''(''θ'')}} satisfies {{EquationNote|4|(2)}}.

== Selected publications ==
*{{Citation
|first=Alfred
|last=Tauber
|title=Über den Zusammenhang des reellen und imaginären Theiles einer Potenzreihe 
|journal=[[Monatshefte für Mathematik und Physik]]
|trans-title=On the relation between real and imaginary part of a power series
|volume=II
|pages=79–118
|year=1891
|url=http://www.literature.at/webinterface/library/ALO-BOOK_V01?objid=12451&page=82&zoom=3&ocr=
|doi=10.1007/bf01691828
|jfm=23.0251.01
|s2cid=120241651
}}.
*{{Citation
 |first        = Alfred
 |last         = Tauber
 |title        = Ueber die Werte einer analytischen Function längs einer Kreislinie
 |trans-title  = On the values of an analytic function along a circular perimeter
 |journal      = [[Jahresbericht der Deutschen Mathematiker-Vereinigung]]
 |volume       = 4
 |page         = 115
 |year         = 1895
 |url          = http://gdz.sub.uni-goettingen.de/index.php?id=11&PPN=PPN37721857X_0004&DMDID=DMDLOG_0041&LOGID=LOG_0041&PHYSID=PHYS_0122&L=1
 |access-date  = 2014-07-16
 |archive-url  = https://web.archive.org/web/20150701131734/http://gdz.sub.uni-goettingen.de/index.php?id=11&PPN=PPN37721857X_0004&DMDID=DMDLOG_0041&LOGID=LOG_0041&PHYSID=PHYS_0122&L=1
 |archive-date = 2015-07-01
 |url-status     = dead
}}.
*{{Citation
|first=Alfred
|last=Tauber
|title=Ein Satz aus der Theorie der unendlichen Reihen
|trans-title=A theorem about infinite series
|journal=[[Monatshefte für Mathematik und Physik]]
|volume=VIII
|year=1897
|pages=273–277
|url=http://www.literature.at/viewer.alo?viewmode=overview&olfullscreen=true&objid=12409&page=280
|doi=10.1007/BF01696278
|jfm=28.0221.02 
|s2cid=120692627
}}.
*{{Citation
|first=Alfred
|last=Tauber
|title=Über einige Sätze der Potentialtheorie
|journal=[[Monatshefte für Mathematik und Physik]]
|trans-title=Some theorems of potential theory
|volume=IX
|pages=79–118
|year=1898
|url=http://www.literature.at/alo?objid=12410
|doi=10.1007/BF01707858
|jfm=29.0654.02
|s2cid=124400762
}}.
*{{Citation
|first=Alfred
|last=Tauber
|title=Über konvergente und asymptotische Darstellung des Integrallogarithmus
|trans-title=On convergent and asymptotic representation of the logarithmic integral function
|journal=[[Mathematische Zeitschrift]]
|volume=8
|pages=52–62
|year=1920
|issue=1–2
|url=http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=GDZPPN002365529
|doi=10.1007/bf01212858
|jfm=47.0329.01
|s2cid=119967249
}}.
*{{Citation
|first=Alfred
|last=Tauber
|title=Über die Umwandlung von Potenzreihen in Kettenbrüche
|trans-title=On the conversion of power series into continued fractions
|journal=[[Mathematische Zeitschrift]]
|volume=15
|pages=66–80
|year=1922
|url=http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=GDZPPN002366959
|doi=10.1007/bf01494383
|jfm=48.0236.01
|s2cid=122501264
}}.

==See also==
*[[Actuarial science]]
*[[Hardy–Littlewood tauberian theorem]]
*[[Summability theory]]

==Notes==
{{reflist|30em}}

==References==
{{refbegin}}

===Biographical and general references===
*{{Citation
|first =Christa
|last =Binder
|author-link =Christa Binder
|editor-last =Chatterji
|editor-first = S. D.
|contribution =Alfred Tauber (1866-1942). Ein österreichischer Mathematiker
|title=Jahrbuch Überblicke Mathematik
|series=Mathematical surveys
|volume=17
|year=1984
|pages=151–166
|language=de
|place =[[Mannheim]]
|publisher =[[Bibliographisches Institut AG]]
|url =https://books.google.com/books?id=qgtWnwEACAAJ
|zbl=0544.01021}}
*{{citation
|editor-first=Gerd 
|editor-last=Fischer
|editor2-first=Friedrich
|editor2-last=Hirzebruch
|editor2-link=Friedrich Hirzebruch
|editor3-first=Winfried 
|editor3-last=Scharlau
|editor4-first=Willi
|editor4-last=Törnig
|title=Ein Jahrhundert Mathematik 1890 – 1990: Festschrift zum Jubiläum der DMV
|place=[[Braunschweig]] / [[Wiesbaden]]
|publisher=[[Vieweg Verlag|Friedrich Vieweg & Sohn]]
|series=Dokumente zur Geschichte der Mathematik
|volume=Band 6
|pages=XII+830
|language=de
|url=https://books.google.com/books?id=nj30BQAAQBAJ
|year=1990
|isbn=3-528-06326-2
|doi=10.1007/978-3-322-80265-1
|mr=1085961
|zbl=0706.01002
}}.
*{{citation
|first1=Maximilian 
|last1=Pinl
|first2=Auguste
|last2=Dick|author2-link=Auguste Dick
|title=Kollegen in einer dunklen Zeit. Schluß
|journal=[[Jahresbericht der Deutschen Mathematiker-Vereinigung]]
|volume=75
|pages=202–203
|language=de
|url=https://eudml.org/doc/146632
|year=1974
|mr=0476359
|zbl=0281.01013
}}.
*{{Citation
|last=Hlawka
|first=Edmund
|author-link=Edmund Hlawka
|contribution=Tauber, Alfred
|contribution-url=http://www.encyclopedia.com/doc/1G2-2830904256.html
|title=[[Complete Dictionary of Scientific Biography]]
|place=New York
|publisher=[[Charles Scribner's Sons]]
|year=2007
|access-date=27 February 2016
}}.
* {{Citation
|last=Sigmund
|first=Karl
|title=Failing Phoenix: Tauber, Helly, and Viennese life insurance
|journal=[[The Mathematical Intelligencer]]
|volume=26
|issue=2
|pages=21–33
|year=2004
|doi=10.1007/bf02985648
|mr=2067894
|zbl=0849.01036
|s2cid=121108996
}}.

===Scientific references===
*{{Citation
|first =S. D.
|last =Chatterji
|editor-last =Chatterji
|editor-first = S. D.
|contribution =Tauber's theorem – a few historical remarks
|title =Jahrbuch Überblicke Mathematik
|series =Mathematical surveys
|volume = 17
|year =1984
|pages =167–175
|place =[[Mannheim]]
|publisher =[[Bibliographisches Institut AG]]
|url =https://books.google.com/books?id=qgtWnwEACAAJ
|zbl=0555.40008
}}, and also {{zbl|0556.01005}}. 
*{{citation
|last=Hardy
|first=G. H.
|author-link=Godfrey Harold Hardy
|title=Divergent Series
|year=1949
|place=[[Oxford]]
|publisher=[[Clarendon Press]]
|lccn=49005496
|oclc=808787
|mr=0030620
|pages=xvi+396
|isbn=978-0-8218-2649-2
|no-pp=true}}, 2nd Edition published by [[Chelsea Publishing Company]], 1991, {{LCCN|91075377}}, {{ISBN|0828403341}}.
*{{Citation
|last1=Hardy
|first1=G. H.
|author-link=Godfrey Harold Hardy
|last2 =Littlewood
|first2 =J. E.
|author2-link =John Edensor Littlewood
|title =Tauberian theorems concerning series of positive terms
|journal =[[Messenger of Mathematics]]
|volume =XLII
|pages =191–192
|year =1913
|jfm =44.0283.01}}.
*{{Citation
|last =King
|first =Frederick W.
|title =Hilbert transforms. Volume 1
|place =[[Cambridge]]
|publisher =[[Cambridge University Press]]
|series =Encyclopedia of Mathematics and its Applications
|volume =124
|year =2009
|pages =xxxviii+858
|isbn =978-0-521-88762-5
|mr =2542214
|zbl =1188.44005
}}.
*{{citation
|first=Jacob
|last=Korevaar
|title=Tauberian theory. A century of developments
|series=Grundlehren der Mathematischen Wissenschaften
|volume=329
|publisher=[[Springer-Verlag]]
|year=2004
|isbn=3-540-21058-X
|pages=xvi+483
|mr=2073637
|zbl=1056.40002
|doi=10.1007/978-3-662-10225-1
}}.
*{{Citation
|last =Lune
|first =J. van de
|title =An introduction to Tauberian theory: from Tauber to Wiener
|place =[[Amsterdam]]
|publisher =[[Centrum Wiskunde & Informatica|CWI]]
|series =CWI Syllabus
|volume =12
|year =1986
|pages =iv+102
|url =http://persistent-identifier.org/?identifier=urn:nbn:nl:ui:18-12912
|isbn =90-6196-309-5
|mr =0882005
|zbl =0636.40002
}}.
{{refend}}

== External links ==
* {{MacTutor Biography|id=Tauber}}
* [http://www.encyclopedia.com/doc/1G2-2830904256.html Alfred Tauber] at encyclopedia.com
* {{MathGenealogy|id=49940}}
{{Authority control}}

{{DEFAULTSORT:Tauber, Alfred}}
[[Category:1866 births]]
[[Category:1942 deaths]]
[[Category:19th-century Hungarian mathematicians]]
[[Category:20th-century Hungarian mathematicians]]
[[Category:Mathematical analysts]]
[[Category:Hungarian people who died in the Theresienstadt Ghetto]]
[[Category:Scientists from Bratislava]]
[[Category:Mathematicians from Austria-Hungary]]
[[Category:University of Vienna alumni]]
[[Category:Academic staff of TU Wien]]
[[Category:Academic staff of the University of Vienna]]
[[Category:Hungarian people executed in Nazi concentration camps]]