Filter design to realize Cauchy product
You could implement some convolutions as recursive filters. Consider moving average as an example, to calculate the output signal, instead of summing over the averaging window, you could add the new input and subtract the part of input which exited the window from the output signal to obtain the output of next time step. Also some filters could be implemented as lattice.

But there is not a general method to implement all filters recursively or as lattice.