Linear L ₁ Approximation for a Discrete Point Set and L ₁ Solutions of Overdetermined Linear Equations

An algorithm for calculating the best linear Li approximation for a discrete point ~t with arbitrary approximating set of functions has been derived. The algorithm handles also the solution of overdetermined linear equations which minimizes the error in the L1 norm. This algorithm is based on a theorem by Hoel, that the polynomials of best pth power approximation converge to the polynomial of best L~ approximation as p --. 1. The coefficients of the /~ approximation are calculated starting with p = 2 and reducing p uniformly from 2 to 1. In any step, the results of the previous step are taken as the initial values for minimizing iteratively the resulting nonlinear equation of the present step. Two numerical examples are given. I.