Shift of unilateral convolution

Theorem

The following formula holds: $$\widehat{(f*g)}(t,s)=\displaystyle\int_s^t \hat{f}(t,\sigma(\xi))\hat{g}(\xi,s) \Delta \xi,$$ where $\widehat{(f*g)}$ denotes the solution of the shifting problem and $(f*g)$ denotes the unilateral convolution.