Summary. We consider the Gauss-Kronrod quadrature formula for the Legendre weight function. On certain spaces of analytic functions its error term is a continuous linear functional. We derive easy to compute estimates for the norm of the error functional, which lead to bounds for the error functional itself. The efficiency of these bounds is illustrated with some numerical examples.

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Abstract.We study Gauss-Kronrod quadrature formulae for the Jacobi weight function «/"'"'(t) = (l-i)Q(l + t)'3 and its special case a = ß = X-^ of the Gegenbauer weight function. We are interested in delineating regions in the (a, /3)-plane, resp. intervals in A, for which the quadrature rule has (a) the interlacing property, i.e., the Gauss nodes and the Kronrod nodes interlace; (b) all nodes contained in (-1,1); (c) all weights positive; (d) only real nodes (not necessarily satisfying (a) and/or (b)). We determine the respective regions numerically for n = 1(1)20(4)40 in the Gegenbauer case, and for n = 1(1)10 in the Jacobi case, where n is the number of Gauss nodes. Algebraic criteria, in particular the vanishing of appropriate resultants and discriminants, are used to determine the boundaries of the regions identifying properties (a) and (d). The regions for properties (b) and (c) are found more directly. A number of conjectures are suggested by the numerical results. Finally, the Gauss-Kronrod formula for the weight w^a'll2^ is obtained from the one for the weight u/a'Q), and similarly, the Gauss-Kronrod formula with an odd number of Gauss nodes for the weight function w(t) = |t|7(l -t2)tt is derived from the Gauss-Kronrod formula for the weight vj(a,(1+1''2\1. Introduction. A Gauss-Kronrod quadrature formula for the (nonnegative) weight function w on [a, b] is a quadrature formula of the form

Abstract. We evaluate explicitly the integrals 1 −1 π n (t)/(r ∓ t)dt, |r| = 1, with the π n being any one of the four Chebyshev polynomials of degree n. These integrals are subsequently used in order to obtain error bounds for interpolatory quadrature formulae with Chebyshev abscissae, when the function to be integrated is analytic in a domain containing [−1, 1] in its interior.

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