# 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.

# Green’s function

The Green’s function is a concept developed by George Green in the XIXth century. It is useful to solve inhomogenous differential equations such as :

$\displaystyle \frac{dy}{dt} = f(t)y(t)+g(t)$

where the functions f and g are defined on an interval.

GREEN’S FUNCTION

More generally, consider an equation with a linear differential operator L :

$\displaystyle Ly = g$

Green’s function

A Green’s function $G(t,s)$ of a linear operator L is any solution of the equation

$\displaystyle LG(t,s) = \delta(t-s)$

where $\delta(t-s)$ is the Dirac delta-function.