I am having some trouble understanding Fourier series rigorously. My first question is, when can we say the fourier series exists? Does it need to converge uniformly? Because we may always just generate the coefficients, but that in itself does not actually say anything about the convergence of the fourier series. So if a question asks to find a fourier series of a function, once I find the the coefficients, is there a test I can do to show that this is indeed the fourier series?
I also have a question on deriving fourier series from other fourier series. For example, deriving the fourier series for $\cos(x)$ from $\sin(x)$, or deriving different the fourier series for different saw-tooth waves' graphs by modifying a fourier series of a normal sawtooth wave.
This, in principle, seems to be like figuring out new power series from old ones, but this feels much harder. I have some "rules of thumb" I follow, like figuring out the period, and modifying the original series accordingly. For example, if our original series was $f(\theta)$, then if the period of our new series is half the period of the old one, then it will be $f(\theta/2)$, and if the range is $\frac{1}{2}$ as big, then it will be $f(\theta)/2$, etc. Are there any other rules I can incorporate to make this easier?
Sorry for the abundance of questions. I want to understand fourier series on a deeper level, and want to clear up any misconceptions.
Thanks again!
There are two approaches for the Fourier series:
One assuming the function is $L^2[0,1]$ and showing $\{e^{2i\pi nx}\}_{n\in \Bbb{Z}}$ is a dense orthonormal family,
The other assuming the function is $C^1$ and expressing $\sum_{n=-N}^N c_n(f)e^{2i\pi nx}$ as $\int_0^1 f(x-y) D_N(y)dy$ where $D_N$ is the Dirichlet kernel, then let $N\to \infty$.
Any of those approaches generalizes to the Fourier series in the sense of distributions which exists for any continuous function as well as all its distributional derivatives.