I have been studying fourier series and fourier transforms and I'm confused about a lot of things mentioned in various questions etc. As an example, I'm a quite puzzled on why fourier transform is used in various applications (signal processing) where we are working with finite functions, and so we might as well assume the signal is a periodic function.
First I repeatedly see people saying "fourier transform of a periodic function" - I don't know what this means. In this question Fourier transform vs Fourier series Lubos Motl says "The Fourier transform of a periodic function is a very special kind of a function, a combination of delta-function". My definition of fourier transform is defined on functions where the integral across $\mathbb{R}$ is well-defined, and I don't see how that is the case for a function that is periodic on $\mathbb{R}$.
Perhaps it means taking a function defined on $(-a,a)$ and extending it by zero on the rest of the domain? In a different answer in the same question this is described.
Second in this question Fourier Transform of a Periodic function
mathreadler answers and shows a picture of a fourier transform of $f(x) = \cos(kx+1)^{24}$, again, I don't understand how you fourier transform that function to get the result in the picture.
Third I can see that a fourier transform won't lose any information for a periodic function. As an example, if we extend a function $f$ defined on $(-\pi, \pi)$ to zero on the rest of the domain, we get that $\frac{1}{2\pi} \hat{f} (\frac{n}{2 \pi})$ gives us the sought fourier coefficient $c_n$ of this function.
However, it seems that we can can explicitly describe the fourier transform simply from the fourier series. It can be shown that $\xi \mapsto \int_{-\pi}^{\pi} e^{i \xi x}$ is a scaled sinc function. Thus, if $f= \sum a_n e^{inx}$, we get that $\hat{f}(\xi) = \sum a_n \int e^{inx - 2\pi i \xi x}$ which is essentially a sum of scaled sinc functions with variously moved origins. So why even work with a fourier transform when we're working in the context of say signal processing, and our signals are defined on an interval of finite measure?