Fourier Series Approximations of Functions

Question

From a few examples of smooth functions, discontinuous functions and continuous functions which have a 'kink' (i.e. $|x|$ where left and right limits disagree)... I've seen that the fourier series much more easily approximates smooth functions and takes significantly more terms to suitably approximate discontinuous functions. This is probably not coincidental, so what is the reason for this?

Answer 1

If $f(x)$ is integrable, then the Fourier series of $f$ is bounded. Now $\widehat{f'}(n) = - i n \hat f(n)$ (or some multiple of this depending on what definition you use). So if $f^{(k)}(x)$ is integrable, then the Fourier coefficients $\hat f(n)$ will be bounded in absolute value by $C/|n|^k$.

In reality this tends to understate how bounded the Fourier coefficients are. For example, the Fourier coefficients of $|x|$ are bounded by something like $C/|n|^2$, whereas what I cited would only give $C/|n|$. This is a huge subject, and I believe much finer results are available. (But I'm not an expert.)