Fourier transform of non-decaying functions

Main Question or Discussion Point

Hi,
if we consider a constant function $f(x)=1$, it is well-known that its Fourier transform is the delta function, in other words:

$$\int_{-\infty}^{+\infty}e^{-i\omega x}dx = \delta(\omega)$$

The constant function does not tend to zero at infinity, so I was wondering: are there other functions that do not tend to zero at infinity but do have a Fourier transform?

I can think only of linear combinations of $$e^{-i\omega x}$$. Are there others?