Consider a function $f_p : \mathbb{R} \to \mathbb{R}$ which is continuously differentiable in $(0,2\pi)$ except at two points $x = x_c$ and $x = x_o$. At $x = x_c$, $f_p(x)$ has a jump discontinuity. At the point $x = x_o$, $f_p'(x_o^+)$ and $f_p'(x_o^-)$ exist and are not equal.
Now define a function $f$ equal to $f_p$ on $(0,2\pi)$ and let it be a periodic function with period $2\pi$. Let $\hat{f}_k$ be the Fourier series coefficients of $f$. What I would like to know is whether the Fourier series defined by the coefficients $ik\hat{f}_k$ converge to $$\frac{f'(x_o^+)+f'(x_o^-)}{2}$$ at $x = x_o$ ?
It will diverge, but if I am not mistaken, can be Cesàro summed.
Your function can be written as a sum of a Lipschitz function that is continuously differentiable except at (at most) two points, and a sawtooth wave. The first component has Fourier series that converges, and has the property that the Fourier series of the derivative is $ik\hat{f}_k$. The second component, however, gives the problem. A sawtooth wave has Fourier series expansion (up to a normalising constant) $\hat{f}_k = \frac{(-1)^k}{k}$. So the main issue is handling the sum of the form
$$ \sum (-1)^k\sin(k x) $$
This of course is not absolutely convergent. But away from the singularity, you can take the Cesàro sum: the partial sum of the above expression is more or less the Dirichlet kernel (shifted by a constant). And so the partial Cesàro sum is (roughly) just the Fejér kernel, which is known to converge point-wise away from the singularity.