in this question, it is explained how one can rigorously define the Fourier transform over $L_1$ functions using a Lebesgue integral. But what about when we transform functions which are not in $L_1$?
Specifically, in many branches of physics and engineering it is common to take Fourier transforms of functions such as $cos(t), \space sin(t),\space f(t)=1$
The above examples all result somehow in Dirac Delta functions, which are actually distributions and not functions proper. So it seems we need to come up with a rather different definition of what a Fourier transform is to include these cases.
How can we do this?