Abstract
The central results of this paper are bounds for the second Peano kernels of the Gaussian quadrature formulae. Hence, for these quadrature formulae, we derive asymptotically optimal error constants for classes of functions with a bounded second derivative or with a first derivative of bounded variation, or for a class of convex functions. To obtain these bounds, we first prove inequalities related to MacMahon's expansion and some further results on the Bessel functionJ o , as well as some “trapezoid theorems for the weights of Gaussian formulae” (cf. Davis and Rabinowitz [6]) with explicit bounds for the error term.
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