The problem of restoration of discrete periodic data on small net via known values on large net is considered. The sum of squares of r-th degree finite differences serves as smoothing functional. It is required to minimize this functional. The explicit algorithm for the solution of the problem is proposed. The algorithm bases on discrete Fourier transform (DFT). The investigation of the solution is carried out as r tends to infinity. The connection with trigonometrical interpolation is established. The results can be used for interpolation of discrete-time signals and periodic spline functions of high degree.