Error bounds for quadrature formulas near Gaussian quadrature

Abstract

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