Carlo Severini

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

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

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, jointly with the obvious requirement that the set ${\displaystyle \{(x,y,z,p)=(0,y,U(y),U^{\mathrm{\prime }}(y));y\in [a,b]\}}$ is contained in the domain of ${\displaystyle f}$.

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 Severini also left an unpublished and unfinished treatise on the theory of real functions, whose title was planned to be "Fondamenti dell'analisi nel campo reale e i suoi sviluppi".