Movatterモバイル変換

Carlo Severini

From Wikipedia, the free encyclopedia

Italian mathematician (1872–1951)

Carlo Severini
Born	10 March 1872 Arcevia (Ancona)
Died	11 May 1951(1951-05-11) (aged 79) Pesaro
Nationality	Italian
Alma mater	Università di Bologna
Known for	Severini-Egorov theorem
Scientific career
Fields	Real analysis
Institutions	Università di Bologna University of Catania University of Genova
Doctoral advisor	Salvatore Pincherle

Carlo Severini (10 March 1872 – 11 May 1951) was anItalian mathematician: he was born inArcevia (Province of Ancona) and died inPesaro. Severini, independently fromDmitri Fyodorovich Egorov, proved and published earlier a proof of the theorem now known asEgorov's theorem.

Biography

[edit]

He graduated inMathematics from theUniversity of Bologna on November 30, 1897:^[1]^[2] the title of his "Laurea"thesis was "Sulla rappresentazione analitica delle funzioni arbitrarie di variabili reali".^[3] After obtaining hisdegree, he worked inBologna as an assistant to the chair ofSalvatore Pincherle until 1900.^[4] From 1900 to 1906, he was a senior high school teacher, first teaching in theInstitute of Technology ofLa Spezia and then in thelyceums ofFoggia and ofTurin;^[5] then, in 1906 he became full professor ofInfinitesimal Calculus at theUniversity of Catania. He worked inCatania until 1918, then he went to theUniversity of Genova, where he stayed until his retirement in 1942.^[5]

Work

[edit]

He authored more than 60 papers, mainly in the areas ofreal analysis,approximation theory andpartial differential equations, according toTricomi (1962). His main contributions belong to the following fields ofmathematics:^[6]

Approximation theory

[edit]

In this field, Severini proved a generalized version of theWeierstrass approximation theorem. Precisely, he extended the original result ofKarl Weierstrass to the class ofbounded locally integrable functions, which is a class including particulardiscontinuous functions as members.^[7]

Measure theory and integration

[edit]

Severini provedEgorov's theorem one year earlier thanDmitri Egorov^[8] in the paper (Severini 1910, pp. 3–4), whose main theme is nevertheless the study ofsequences oforthogonal functions and their properties.^[9]

Partial differential equations

[edit]

Severini proved anexistence theorem for theCauchy problem for thenon linear hyperbolic partial differential equation of first order

\left\{{\begin{array}{lc}{\frac {\partial u}{\partial x}}=f\left(x,y,u,{\frac {\partial u}{\partial y}}\right)&(x,y)\in \mathbb {R} ^{+}\times [a,b]\\u(0,y)=U(y)&y\in [a,b]\Subset \mathbb {R} \end{array}}\right.,

assuming that the Cauchy data $U {\displaystyle U}$ (defined in thebounded interval $[a,b]$ ) and that thefunction $f {\displaystyle f}$ hasLipschitz continuous first orderpartial derivatives,^[10] jointly with the obvious requirement that theset $\{(x,y,z,p)=(0,y,U(y),U^{\prime }(y));y\in [a,b]\}$ is contained in thedomain of $f {\displaystyle f}$ .^[11]

Real analysis and unfinished works

[edit]

According toStraneo (1952, p. 99), he worked also on the foundations of the theory ofreal functions.^[12] Severini also left an unpublished and unfinishedtreatise on the theory ofreal functions, whose title was planned to be "Fondamenti dell'analisi nel campo reale e i suoi sviluppi".^[13]

Selected publications

[edit]

Severini, Carlo (1897) [1897-1898],"Sulla rappresentazione analitica delle funzioni reali discontinue di variabile reale" [On the analytic representation of discontinuous real functions of a real variable],Atti della Reale Accademia delle Scienze di Torino. (in Italian),33:1002–1023,JFM 29.0354.02. In this paper Severini extends the standard Weierstrass approximation theorem to a wider class of functions characterised by the fact that they can have a particular kind of discontinuities.
Severini, C. (1910),"Sulle successioni di funzioni ortogonali" [On sequences of orthogonal functions],Atti dell'Accademia Gioenia, serie 5^a (in Italian),3 (5): Memoria XIII,1–7,JFM 41.0475.04. This paper contains Severini's most known and cited result, i.e. the Severini–Egorov theorem.

Notes

[edit]

^According to the summary of his student file available from theArchivio Storico dell'Università di Bologna (2004) (an electronic version of thearchives of theUniversity of Bologna).
^The content of this section is based on references (Tricomi 1962) and (Straneo 1952): this last one also refers that he was married and had several children, however without giving any other detail.
^AnEnglish translation reads as "On the Analytic Representation of Arbitrary Functions of Real variables"; despite the similarities in the title and the same year of publication, the biographical sources do not say if the paper (Severini 1897) is somewhat related to his thesis.
^The1897–1898 yearbook of the university already lists him between theassistant professors.
^^a ^bAccording toStraneo (1952, p. 98).
^Only his most known results are described in the following sections:Straneo (1952) reviews his research in greater detail.
^According toStraneo (1952), the result is given in various papers, source (Severini 1897) perhaps being the most accessible of them.
^Egorov's proof is given in the paper (Egoroff 1911).
^Moreover, according toStraneo (1952, p. 101), while acknowledging his own priority in the publication of the result, Severini was unwilling to disclose it publicly: it wasLeonida Tonelli who, in the note (Tonelli 1924), publicly credited him the priority for the first time.
^This means that f belongs to theclass $C^{(1,1)}$ .
^For more details about his researches in this field, see (Cinquini-Cibrario & Cinquini 1964) and the references cited therein
^Straneo (1952, p. 99) lists Severini's researches on this field under as "Fondamenti dell'analisi infinitesimale (Foundations of infinitesimal analysis)": however, the topics covered range from the theory of integration toabsolutely continuous functions and to operations on series of real functions.
^"Foundations of Analysis on the Real Field and its Developments": again according toStraneo (1952, p. 101), the treatise would have included his later original results and covered all the fundamental topics required for the study offunctional analysis on thereal field.

References

[edit]

Biographical and general references

[edit]

Archivio Storico dell'Università di Bologna (2004) [1897],"Carlo Severini",Fascicoli degli studenti, Fascicolo della Facoltà di Scienze Fisiche Matematiche Naturali n° (in Italian),2843, archived fromthe original on March 10, 2012, retrievedMarch 1, 2011. A very short summary of the student file of Carlo Severini, giving however useful information about hislaurea.
Straneo, Paolo (1952),"Carlo Severini",Bollettino della Unione Matematica Italiana, Serie 3 (in Italian),7 (3):98–101,MR 0050531, available from theBiblioteca Digitale Italiana di Matematica. Theobituary of Carlo Severini.
Tonelli, Leonida (1924),"Su una proposizione fondamentale dell'analisi" [On a fundamental proposition of analysis],Bollettino della Unione Matematica Italiana, Serie 2 (in Italian),3:103–104,JFM 50.0192.01. In this short note Leonida Tonelli credits Severini for the first proof of Severini–Egorov theorem.
Tricomi, F. G. (1962),"Carlo Severini",Matematici italiani del primo secolo dello stato unitario,Memorie dell'Accademia delle Scienze di Torino. Classe di Scienze fisiche matematiche e naturali. Serie IV (in Italian), vol. I, Torino, p. 120,Zbl 0132.24405, archived fromthe original on 2011-01-11, retrieved2010-05-21{{citation}}: CS1 maint: location missing publisher (link). "Italian mathematicians of the first century of the unitary state" is an important historical memoir giving brief biographies of the Italian mathematicians who worked and lived between 1861 and 1961. Its content is available from the website of the .
Università di Bologna (1898),"Facoltà di Scienze Fisiche, Matematiche e Naturali. Assistenti",Annuario della Regia Università di Bologna (in Italian), Bologna: Premiato Stabilimento Tipografico Succ. Monti, p. 170.