I understand that the Fourier Series for a discontinuous function, one with a jump discontinuity such as the sawtooth wave, converges to the average of the left and right limits at the discontinuity.
Is there some way by which the left and right limits themselves can be read out from the Fourier series representation?
That is, a way to write these quantities $$\lim_{x\rightarrow x_0^+} \sum_{n=-\infty}^{\infty} c_n \exp(i n x)$$ $$\lim_{x\rightarrow x_0^-} \sum_{n=-\infty}^{\infty} c_n \exp(i n x)$$ directly as series over $c_n$, without the limits.
When you have a single jump discontinuity then you can work numerically. Say, you have a periodic function $f$ which is smooth, apart from a jump at the points $2k\pi$. Let $$J(t):={\pi-t\over2}\quad(0<t<2\pi),\qquad J(t+2\pi)\equiv J(t)\ ,$$ be your favorite jump function. Then $$J(t)\rightsquigarrow\sum_{k=1}^\infty{\sin(kt)\over k}\ .$$ The auxiliary function $$g(t):=f(t)-c J(t)$$ is then everywhere smooth for a suitable choice of $c$, hence has Fourier coefficients going to $0$ much more rapidly than ${1\over k}$. It follows that the Fourier sine coefficients of $f$ decay as fast as $b_k\approx{c\over k}$ $(k\gg1)$. This allows to obtain an estimate for $c$, maybe doing some minimization work involving many coefficients. The jump height then is $\pi c$.