# 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.

## Biography

He graduated in Mathematics from the University of Bologna on November 30, 1897: the title of his "Laurea" thesis was "Sulla rappresentazione analitica delle funzioni arbitrarie di variabili reali". After obtaining his degree, he worked in Bologna as an assistant to the chair of Salvatore Pincherle until 1900. 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; 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.

## 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:

### 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.

### Measure theory and integration

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

### Partial differential equations

Severini proved an existence theorem for the Cauchy problem for the non 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 $U$ (defined in the bounded interval $[a,b]$) and that the function $f$ has Lipschitz continuous first order partial derivatives, jointly with the obvious requirement that the set $\scriptstyle \{(x,y,z,p)=(0,y,U(y),U^{\prime }(y));y\in [a,b]\}$ is contained in the domain of $f$.

### Real analysis and unfinished works

According to Straneo (1952, p. 99), he worked also on the foundations of the theory of real functions. 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".

## 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.