An algorithm for the H^∞ optimal filter design is presented for linear systems with pure delays in state and measurement. The problem is reduced to a single linear functional equation. The optimal filter, as well as the optimal regulator in the dual control problem, are obtained in term of a solution to this equation without additional computations. The duality transformation is presented for systems in the input-output form. The pure interpolation problem, where an analytical solution can be obtained from a polynomial factorization equation, is studied as a particular case. The approach presented provides a numerical algorithm for delayed systems of a general type.