Error bounds for quadrature formulas near Gaussian quadrature

摘要

Let Rn be the error functional of a quadrature formula Qn on [−1,1] using n nodes. In this paper we consider estimates of the form |Rn[ƒ]|⩽cm∥ƒ(m)∥, ∥ƒ∥≔sup|x|⩽1|ƒ(x)|, with best possible constant cm, i.e., cm = cm(Rn)≔ sup∥ƒ(m)∥⩽1|Rn[ƒ]|. For the error constants c2n−k(RGn) of the Gaussian quadrature formulas QGn we prove results, which are asymptotically sharp, when n increases and k is fixed. For this latter case, comparing with the corresponding error constants c2n−k(Rn) of every other quadrature formula Qn, we show that the order of magnitude of c2n−k(RGn) cannot be improved in n. In particular, we investigate the question of minimal and maximal values of c2n−k(Rn) in the class of all quadrature formulas Qn having at least algebraic degree of exactness deg(Qn)⩾2n−k−1.