Suppose I have a function $s(t)$ that is convoluted with a set of kernels $k(t,\tau)$ for $t>0$. Let the kernels be a square pulse of length $\tau$. Given I know the convolution \begin{equation} m(t,\tau) = \int s(t')\, k(t'-t,\tau) \,\,\mathrm{d}t' \end{equation} for different values of $t$, how can I reconstruct $s(t)$?
If I have $m(t,\tau=\tau_1)$ for one value $\tau_1$, this would be a deconvolution (numerically done e.g. via a Wiener deconvolution). If I have $m(t=0,\tau)$, this would be a differentiation with respect to $\tau$ ($\mathrm{d}/\mathrm{d}\tau$). It there a generalization to reconstruct $s(t)$ in case I know it for multiple combinations of $t$ and $\tau$? In the convolution picture, this would be the same function $s(t)$ being convoluted with a set of kernels $k(t,\tau_i)$.