On the “Best” and “Least Qth” Approximation of an Overdetermined System of Linear Equations

There has been a renewed interest lately in the computational problem of constructing "best-approximations" of continuous functions of one variable by other continuous functions of one variable. Several methods [1, 2, 3] have already been discussed and many examples of approximating functions of one variable by polynomials and rational functions, etc. have been published. On the other hand the important problem of solving an overdetermined system of linear equations in the Tchebycheff sense has not been given the attention it deserves. The possession of a convenient method for the solution of this problem would make practical "best-approximations" of functions of several variables by linear combinations of arbitrary functions, in much the same manner as this is done by the method of "least squares". It is not widely knox~]l that an algorithm for solving this problem exists and is due to de la Vall~e Poussin [4]. Although the algorithm is unwieldy in use it is important for guiding the computation of a "best-approximation". We shall therefore outline the results of de la ValiSe Poussin before discussing some new methods with which we have been experimenting.