Summary: A specific isomorphism is constructed via the transform domains between the analog signal space \(L^2(-\infty, \infty)\) and the digital signal space \(l_2\). It is then shown that the class of linear time-invariant realizable filters is invariant under this isomorphism, thus demonstrating that the theories of processing signals with such filters are identical in the digital and analog cases. This means that optimization problems involving linear time-invariant realizable filters and quadratic cost functions are equivalent in the discrete-time and the continuous-time cases, for both deterministic and random signals. Finally, applications to the approximation problem for digital filters are discussed.