New Error Coefficients for Estimating Quadrature Errors for Analytic Functions

Since properly normalized Chebyshev polynomials of the first kind T„(z) satisfy (?„, ?„) = [ f,(z)7ÜÖ |1 z2\~TM\dz\ = Smn for ellipses ep with foci at ± 1, any function analytic in ep has an expansion,/(z) = J3 anfn{z) with a„ = (/, Tn). Applying the integration error operator E yields E(J) = 2~Z a„E(Tn)Applying the Cauchy-Schwarz inequality to E(J) leads to the inequality |£CDIsá Z W¿2\E(Tn)\2 = \\fm\E\\2. \\E\\ can be computed for any integration rule and approximated quite accurately for Gaussian integration rules. The bound for \E(f)\ using this norm is compared to that using a previously studied norm based on Chebyshev polynomials of the second kind and is shown to be superior in practical situations. Other results involving the latter norm are carried over to the new norm. 1. Davis and Rabinowitz [6], following the work of Davis [3], developed a new method for bounding the truncation error in the numerical integration of functions analytic over the interval of integration, standardized to [—1, 1]. This method was based on the fact that every such function could be continued analytically into a region enclosed by one of a family of confocal ellipses ep, with foci at ±1, where p — a + b, a is the semimajor axis of ep, and b = (o2 — 1)1/2 is the semiminor axis. Error coefficients &ÍR, p) were computed for various values of p and for several integration rules R, where (1) Rif) = ¿ H-./O0 ¿-i is determined by a particular choice of weights w¿ and abscissas xt, i = 1, • • • , n. The ciR, f) were computed using the Chebyshev polynomials of the second kind UJiz) which are orthogonal over the interior of e„ with respect to the inner product (2) (/, g)'P = ff f{z)g~iz) dx dy. They are given explicitly by t'e formula . . ik + öl"1 tt!)k È "«i«*)T /,^ 2,p , 4 ^ L k + 1 frl_J_ (3) a iR, p) = ¿_,-21+2-=2t=2^_ T 4-0 P — P Received November 26, 1968, revised December 18, 1969. AMS Subject Classifications. Primary 6555, 6580.