2 Existence of Fourier transforms

Formally, sufficient conditions for the Fourier transform of a function $f\left(t\right)$ to exist are

1. ${\int \; <!--nolimits\; -->}_{-\infty }^{\infty }{\left|f\left(t\right)\right|}^{2}dt$ is finite
2. $f\left(t\right)$ has a finite number of maxima and minima in any finite interval
3. $f\left(t\right)$ has a finite number of discontinuities.

Like the equivalent conditions for the existence of Fourier series these conditions are known as Dirichlet conditions .

If the above conditions hold then $f\left(t\right)$ has a unique Fourier transform. However certain functions, such as the unit step function, which violate one or more of the Dirichlet conditions still have Fourier transforms in a more generalized sense as we shall see shortly.