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.

# Integral Kernel

# 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 :

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 :

**Green’s function**

A Green’s function of a linear operator L is any solution of the equation

where is the Dirac delta-function.