I understand the easy proof of Gibb's Phenomenon where you first prove it for the (periodically extended) square wave $f(t) = \begin{cases} c & 0<t<1 \\ -c & -1 < t < 0 \end{cases}$ for some constant $c$ and then use this to write a general "nice enough" function with jump discontinuity as a sum of a function whose Fourier series uniformly converges and a finite sum of square waves.
The question is then whether there's a $\textit{direct}$ proof of Gibb's phenomenon in the general jump discontinuity case that doesn't revert to the square wave case. I understand that the very explicit nature of the square wave case makes such a proof likely unnecessarily complicated compared to the proof above, but still this is quite interesting of a question I think.
