In this paper we give an alternative and, in our opinion, more simple proof for the orthonormal discrete polynomials on a set of equidistant nodes. Such a proof provides a unifying explicit formulation of discrete orthonormal polynomials on an equidistant grid and an explicit formula for the coefficients of the “three-term recurrence relation”. We show how to efficiently apply these formulas to the problem of least-squares fitting on equidistant nodes. Finally we investigate the problem to determine the effect of adding a new basis function or taking one away. This is useful in the process of trying to discover the optimal set of basis functions and the problem to update them when a moving window over the time series/data stream is used. Mathematics Subject Classification: 05E35, 93E24