 Solution

Not as ordinary convergent Fourier series, for a nontrivial moving ramp. Formal spatial differentiation would giveand formal time differentiation multiplies that expression by $-v$. The summands have nondecaying amplitude and do not tend to zero, so the ordinary series cannot converge. The periodic extension has jumps, violating the regularity needed for Termwise differentiation of a Fourier series.

Away from the moving jumps, the classical derivatives exist and are $g_x=K/L$ and $g_t=-vK/L$. A distributional derivative also includes the Dirac delta distribution contributions at the jumps, so distributional differentiation is possible, but it is a different interpretation. If $v=0$, the time derivative is trivially zero; if $K=0$, both derivatives are zero.