A digital-computer formulation of the delay-line-synthesiser technique is developed, and the need for pre- and postfiltering to avoid aliasing effects is emphasised. This leads naturally to Fourier-series approximation and the least-squares error criterion. Special attention is given to numerical methods of calculating the optimum set of weights for the delay-line synthesiser, ρfactors are proposed for smoothing the approximation in the frequency domain; they appear to have a slight computational advantage, compared with the σfactors of Lanczos. A hybrid-computer method is proposed for solving the finite trigonometric-series expression for the optimum weights. The proposed method almost halves the number of arithmetic operations required by numerical analysis alone.