# 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.

# Division by increasing power order

Euclid conducting a proof (Raphael's fresco School Of Athens, 1511)

The division of polynomials by increasing power order is similar to the usual Euclidean division but in reversed order : the terms of lower degree of the dividend are eliminated first. Some useful algorithms can be deduced from this process, such as series expansion or partial fraction decomposition.

EUCLIDEAN DIVISION

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

$\displaystyle A = B Q + R , \deg(R)<\deg(B)$