The evaluation and estimation of the coefficients in the Chebyshev series expansion of a function

Recent investigations have considered the application of Chebyshev series to finding numerical solutions to frequently occurring problems. The quadrature problem has been considered by Clenshaw and Curtis [1]; the numerical solution of ordinary differential equations by Clenshaw [2], Fox [3] and Clenshaw and Norton [4]; the numerical solution of Fredholm integral equations by the author [5] and [6]; and finally a simple partial differential equation by the author [7]. In many of these applications it is useful to be able to estimate the degree N of the polynomial approximation to a given function f(x), so that f(x) is then represented to within some prescribed accuracy. In order to do this some knowledge of the magnitude of the coefficients an is required, in general for large values of n. An attempt has been made to do this by the author [6], but the estimates given there are in general fairly conservative. As an example, it is shown that if f(x) is infinitely differentiable in -1 ? x < 1 with I f(n) (X) < Qn then