On the Gaussian Integration of Chebyshev Polynomials

It is shown that as m tends to infinity, the error in the integration of the Chebyshev polynomial of the first kind, T{im+2)j±2^x), by an /n-point Gauss integration rule approaches (-!> • 2/(4/2 1), / = 0, 1, ■ • • , m 1, and (-!>' • tt/2, / = m, for all J. 1. Knowledge of the errors in the numerical integration of Chebyshev polynomials of the first kind, Tn(x), by given integration rules has proved to be useful in various situations. On the one hand, they can be used in estimating the error in integrating functions of low-order continuity [4] or with branch-point singularities [5]. On the other hand, they are needed in computing the norm of the error functional of the given rule in a certain family of Hilbert spaces of analytic functions [7]. For certain rules, namely for Gauss, Lobatto and Radau rules, asymptotic, and in some cases exact, values of these errors for certain values of the parameters involved were given in [3]. In this paper, we give further asymptotic results for the case of Gauss integration which are valid for all values of the parameters, thus completing the picture in this particular but very important case. These asymptotic results, which agree with the true computed results quite early, have been used to explain why the use of a Gauss rule with an even number of points, say In, is superior to one with an odd number of points, 2n + 1, in integrating a function which is not analytic at the midpoint of the integration interval [6]. Numerical evidence suggests that similar results hold for a second family of important rules, the Lobatto rules, but the tools at our disposal in this case are not as powerful as in the well-investigated Gauss case and, hence, we were unable to prove these results. 2. The Chebyshev polynomials of the first kind, Tn(x), are defined in I = [— 1, 1] by Tn(x) = cos nd, where x = cos 0, 0 ;£ 0 — it. We shall denote the error in integrating T„(x), using a Gauss m-point rule, by /l m Tn(x)dx £ N>,ltt*) •1 i-l where xt = xi-m, i = 1, • • • , m, are the zeros of the Legendre polynomial of degree m and Wt = witm, i = I, • ■ • , m, are the corresponding weights which are all positive with E"-i w< = 2. Since Tn(x) is odd for n odd and since the Gauss rules are symmetric about the origin, it follows that Em(T2k+1) = 0 for all k. Furthermore, an m-point Gauss rule is exact for all polynomials of degree :S2m — 1. Hence, we shall only consider Em(T2k), where k m. In this case, we have Received March 12, 1971, revised June 6, 1971. AMS 1970 subject classifications. Primary 65D30; Secondary 33A65.