General theory of approximation by functions involving a given number of arbitrary parameters

Abstract

The following is a special case of the problem to be considered in this paper : Given a function cb(x) of the real variable x, continuous on a finite interval (a, b) ; to determine the polynomial p(x) of given degree n, which gives the closest approximation to the given function cp on the interval (a, b).This problem becomes definite only when the meaning of the phrase " closest approximation " has been precisely stated, and the meaning adopted will depend on the ultimate object in view.Tchebtchev seems to have been the first to consider this problem.tHe regarded that polynomial as giving the best approximation, which rendered the maximum of \p(x) -cf>(x)\, &a x varied over (a, b), as small as possible.A different point of view would lead one to seek a polynomial of the given degree which rendered as small as possible the expression C(p-eb)2dx,Ja or the expression j \p -eb\dx, etc.In all of these cases, and in the more general ones to be referred to presently, the problem consists in the determination of a set of parameters a. of a function f(x; a0, ax, -, af) of the real variable x, such that the maximum of |y(x)|, as x varies over a given interval, shall be as small as possible.