In this paper, univariate cubic L
1 interpolating splines based on the first derivative and on 5‐point windows are introduced. Analytical results for minimizing the local spline functional on 5‐point windows are presented and, based on these results, an efficient algorithm for calculating the spline coefficients is set up. It is shown that cubic L
1 splines based on the first derivative and on 5‐point windows preserve linearity of the original data and avoid extraneous oscillation. Computational examples, including comparison with first‐derivative‐based cubic L
1 splines calculated by a primal affine algorithm and with second‐derivative‐based cubic L
1 splines, show the advantages of the first‐derivative‐based cubic L
1 splines calculated by the new algorithm.
