I'm trying to detect rapid changes in a one-dimensional signal say $[0,1]\ni t \mapsto f(t) \in [-1,1]$. By rapid changes, I mean corner points, edges, or sharp transitions at a point for example the signal switches from a value $-1$ to $+1$ on a point.
For this, I'm using the STFT i.e., the Short-Time Fourier transform using some window functions (see Foundations of Time-Frequency Analysis by Grochenig, Chapter 3).
My question is:
Is there any justification or estimate indicating for what classes of signals STFT can provide some kind of qualitative answer about the discontinuity detection accuracy? I see that mathematically STFT can be applied for any $L^2([0,1];[-1,1])$ class of signals which can be very wild. But how does the data derived from an STFT, i.e., the time-frequency plot indicates, mathematically which part of the signal is jittery or at which point the signal switches? Is there any formal results/theorems in this line?
