I am taking a signals and systems class and within it is the idea of taking a Fourier transform of a periodic function.
Can someone explain this and put it on rigorous footing? Clearly such a function doesn't exist, even as a limit. The concept of "distribution" doesn't really apply here either as it is being treated as a function. For example it is being multiplied with a LTI frequency-domain transfer function then inverted to get the output of the system to the periodic input.
What is going on?
The Fourier transform of a $1$-periodic function $f(t)$ is the distribution
$$\hat{f} (\xi)=\sum_{n=-\infty}^\infty \delta(\xi−n)c_n$$ where $c_n= \int_0^1 f(t)e^{−2i\pi nt}dt$ are the Fourier series coefficients.
It is a major difficulty in signal processing : for unifying
the discrete Fourier transform,
the Fourier series,
the discrete-time Fourier transform,
and the Fourier transform
you need the Fourier transform of distributions.