I've heard two heuristics for checking the uniform convergence of a Fourier series, but I am missing details and verification.
The first involves the Fourier series (or is it the original function?) having gaps. If the Fourier series has gaps (does this include jump discontinuities?) then it is not uniform convergent to the original function. This one confuses me as I've heard it can be piecewise continuous and still be convergent. I've also heard this as $F$ must be continuous.
The second heuristic involves the decay of $a_n$ and $b_n$; they have to decay faster than $1/n$? Or is it $1/n^2$? Can't this happen if the series has jump discontinuities/gaps?
Thank you for anyone who can fill in the details for me.