We consider the estimation of the parameter of the best interpolator from a restricted class of linear, time-invariant interpolators for the weakly stationary process {xt}. The estimation is supposed to be based on a part-realisation x1, · ··, xn and on certain prior information regarding the spectral density of the process. Two methods of estimation are proposed — one which uses x1, · ··, xn and the prior information and another which uses only x1, · ··, xn. For the methods proposed it is reasonable to conjecture that the method which utilises the prior information as well as x1, · ··, xn, will be more efficient. Reasonable as this conjecture may seem, its proof is by no means obvious and is the main subject of the paper.
