Truncation Errors in Two Chebyshev Series Approximations

However, in attempting to find a suitable polynomial approximation to a general function f(x), the integral occurring in equation (1.3) cannot be evaluated explicitly, and recourse has to be made to approximate methods for evaluating an. The most widely used method is the "curve-fitting" method described by Lanczos [1], and, in greater detail, by Clenshaw and Curtis [2]. There are two variations of the method which we shall call the "practical" and "classical" methods, respec, tively. Suppose we wish to approximate to f(x) by a polynomial of degree N. In the "practical" method, we construct a polynomial IN(x) by collocation with f(x) at the (N + 1) points xi = cos (7ri/N), i = 0(1 )N, which are the zeros of the polynomial [TN+1(X) TN_1(x)]. In the "classical" method, we construct a polynomialNd(x) by collocation with f(x) at the (N + 1) points