The determination of the Chebyshev approximating polynomial for a differentiable function

Abstract

generality, to be the interval -1 < x < 1, there exists a unique polynomial Pn *(x) of given maximum degree n, which is such that the maximum of I f(x)- P.*(x) I over-I < x < ? is less than the maximum of if(x) - P.(x) I over-i < x < 1, where Pn(x) is any other polynomial of degree not exceeding n. The polynomial Pn*(x) is known as the Chebyshev approximation, of maximum degree n, to f(x) over-1 < x < 1. It is characterized by the fact that f(x) - Pn*(x) assumes extreme values at n + 2 points, at least, of the interval - 1 < x < 1, these extreme values being equal in magnitude and alternating in sign [1]. We refer to the points of any such set of n + 2 points as critical points, and we denote them by (xi* , X*+2), where -1 < xi* < < x*+2 < 1. Thus, the end points, +1, of the interval -1 < x ? 1 may be critical points, but at least n of the n + 2 critical points, namely, x2*, , x +i, are interior points of this interval. We assume that f(x) is not only continuous, but also differentiable, over -1 ? x < 1, and so the derivative of f(x) - Pn*(x) is zero at each of the n points x2*, * n+ If the derivative of f(x) - Pn*(x) cannot be zero more than n times, it follows that xi* = 1, Xn+2 = 1 and that the derivative of f(x) -Pn*(x) has precisely n zeros that are interior points of the interval -1 < x < 1. The polynomial Pn*(x) is an odd function of x when f(x) is odd, and is an even function of x when f(x) is even. Thus, when f(x) is odd we may take n to be even, the maximum degree of Pn*(x) being n - 1, and when f(x) is even we may take n to be odd, the maximum degree of Pn*(x) being again n - 1. In these cases the critical points are distributed symmetrically about the mid-point x = 0 of the interval -1 ? x < 1, and we may confine our attention to the part 0 < x < 1 of this interval. When f(x) is odd, so that n is even, the number of critical points is even and x = 0 is not a critical point; on the other hand, when f(x) is even, the number of critical points is odd and x = 0 is a critical point. When f(x) is odd, or even, and x = 1 is a critical point, we change our notation and denote the positive interior critical points by xi < X2* < ... < Xk*, where n = 2k in the first case, and n = 2k + 1 in the second. For example, when f(x) = arc tan x, P2*(x) is an odd polynomial of degree ? 2k - 1, and so the derivative of arc tan x - P2*k(x) cannot vanish more than 2k times; this implies that the points ?41 are critical points, and, in addition, since this derivative must vanish 2k times, that P2* (x) is