Carlo Severini

Carlo Severini (10 March 1872 – 11 May 1951) was an Italian mathematician: he was born in Arcevia (Province of Ancona) and died in Pesaro. Severini, independently from Dmitri Fyodorovich Egorov, proved and published earlier a proof of the theorem now known as Egorov's theorem.

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

Biography

He graduated in Mathematics from the University 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 his degree, he worked in Bologna as an assistant to the chair of Salvatore Pincherle until 1900.^[4] From 1900 to 1906, he was a senior high school teacher, first teaching in the Institute of Technology of La Spezia and then in the lyceums of Foggia and of Turin;^[5] then, in 1906 he became full professor of Infinitesimal Calculus at the University of Catania. He worked in Catania until 1918, then he went to the University of Genova, where he stayed until his retirement in 1942.^[5]

Work

He authored more than 60 papers, mainly in the areas of real analysis, approximation theory and partial differential equations, according to Tricomi (1962). His main contributions belong to the following fields of mathematics:^[6]

Approximation theory

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

Measure theory and integration

Severini proved Egorov's theorem one year earlier than Dmitri Egorov^[8] in the paper (Severini 1910), whose main theme is however sequences of orthogonal functions and their properties.^[9]

Partial differential equations

Severini proved an existence theorem for the Cauchy problem for the non linear hyperbolic partial differential equation of first order

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

assuming that the Cauchy data ${\displaystyle U}$ (defined in the bounded interval ${\displaystyle [a,b]}$) and that the function ${\displaystyle f}$ has Lipschitz continuous first order partial derivatives,^[10] jointly with the obvious requirement that the set ${\displaystyle \scriptstyle \{(x,y,z,p)=(0,y,U(y),U^{\prime }(y));y\in [a,b]\}}$ is contained in the domain of ${\displaystyle f}$.^[11]

Real analysis and unfinished works

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

Selected publications

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

See also

• Hyperbolic partial differential equation
• Orthogonal functions
• Severini-Egorov theorem
• Weierstrass approximation theorem

Notes

1. According to the summary of his student file available from the Archivio Storico dell'Università di Bologna (2004) (an electronic version of the archives of the University of Bologna).
2. 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.
3. An English 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.
4. The 1897–1898 yearbook of the university already lists him between the assistant professors.
5. According to Straneo (1952, p. 98).
6. Only his most known results are described in the following sections: Straneo (1952) reviews his research in greater detail.
7. According to Straneo (1952), the result is given in various papers, source (Severini 1897) perhaps being the most accessible of them.
8. Egorov's proof is given in the paper (Egoroff 1911).
9. Also, according to Straneo (1952, p. 101), Severini, while acknowledging his own priority in the publication of the result, was unwilling to disclose it publicly: it was Leonida Tonelli who, in the note (Tonelli 1924), credited him the priority for the first time.
10. This means that f belongs to the class ${\displaystyle C^{(1,1)}}$.
11. For more details about his researches in this field, see (Cinquini-Cibrario & Cinquini 1964) and the references cited therein
12. 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 to absolutely continuous functions and to operations on series of real functions.
13. "Foundations of Analysis on the Real Field and its Developments": again according to Straneo (1952, p. 101), the treatise would have included his later original results and covered all the fundamental topics required for the study of functional analysis on the real field.

References

Biographical and general references

• 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 from the original on March 10, 2012, retrieved March 1, 2011. A very short summary of the student file of Carlo Severini, giving however useful information about his laurea.
• Straneo, Paolo (1952), "Carlo Severini", Bollettino della Unione Matematica Italiana, Serie 3 (in Italian), 7 (3): 98–101, MR 0050531, available from the Biblioteca Digitale Italiana di Matematica. The obituary 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 from the original on 2011-01-11, retrieved 2010-05-21{{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.

Scientific references

• Cinquini-Cibrario, M.; Cinquini, S. (1964), Equazioni a derivate parziali di tipo iperbolico [Partial differential equations of hyperbolic type], Monografie matematiche del Consiglio Nazionale delle Ricerche (in Italian), vol. 12, Roma: Edizioni Cremonese, pp. VIII+552, MR 0203199, Zbl 0145.35404. A monograph surveying the theory of hyperbolic equations up to its state of the art in the early 1960s, published by the Consiglio Nazionale delle Ricerche.
• Egoroff, D. Th. (1911), "Sur les suites des fonctions mesurables", Comptes rendus hebdomadaires des séances de l'Académie des sciences (in French), 152: 244–246, JFM 42.0423.01, available at Gallica.

External links

• Guerraggio, Angelo; Nastasi, Pietro; Tricomi, Francesco (2008–2010), Carlo Severini (1872 – 1951) (in Italian), retrieved March 2, 2011. Available from the Edizione Nazionale Mathematica Italiana.