# Volterra Series

Vito Volterra (1860-1940) was one of the founding fathers of functional analysis. At the turn of the twentieth century, he introduced the notion of functions of lines, which are defined over a functional space. He studied their derivative and obtained results on integral equations. Practically, a Volterra series is a polynomial functional expansion similar to a Taylor series that provides an approximation of weakly nonlinear systems. One of the first application to nonlinear system analysis is due to Wiener in the 1940s, who developed a method for determining the nonlinear response to a white noise input. Nowadays, this approach is widely used for system identification in many domains such as electrical engineering or biological sciences.

It is well known that given two polynomials $A$ and $B \neq 0$ on a commutative field K, it is always possible to perform the Euclidean division
$\displaystyle A = B Q + R , \deg(R)<\deg(B)$