As a toy question, consider the Fourier sine series of the function $1_{[a,b]}$ on $[0,\pi]$. We know the point-wise convergence of $$f(x)=1_{[a,b]}(x) = \frac{2}{\pi}\sum_{n=1}^\infty \frac{\cos(na)-\cos{nb}}{n}\sin(nx).$$ Of course this function is not differentiable, but it is piecewise smooth and term-by-term differentiation makes sense that $$f'(x) = 0\quad ``=''\quad \frac{2}{\pi}\sum_{n=1}^\infty \left(\cos(na)-\cos(nb)\right)\cos(nx)$$ if $x\ne a, b$. These sums are not absolutely convergent, but it's okay in the Cesàro sum sense.
Now assuming we don't know what was the functions on LHS above and just have the Fourier series (for example as a weak solution of some equation). Is there any result of convergence for this case? To clarify, say we know the Cesàro summable Fourier series $S(x)$ for some unknown function $f(x)$. It is supposed that $f'(x)$ blows up at some singularities where the term-by-term differentiation $S'(x)$ diverges in the Cesàro sense. Is there any general theorem I can use when I want to say $S'(x)=f'(x)$ away from those singularities?
I think I may use bump functions to have such result in the specific case I'm interested in, but it would be great if there was a general result on it.