Asymptotically optimal error bounds for quadrature rules of given degree

Abstract

If the quadrature rule Q is applied to the function f, then the error can in many situations be bounded by ρ s ( Q ) ‖ f ( s ) ‖ ∞ {\rho _s}(Q){\left \| {{f^{(s)}}} \right \|_\infty } , where ρ s ( Q ) {\rho _s}(Q) is independent of f. We obtain the asymptotics of these numbers for the Gaussian method Q n G ( n = 1 , 2 , … ) Q_n^{\text {G}}\;(n = 1,2, \ldots ) with very general weight functions and show that ρ s ( Q n G ) {\rho _s}(Q_n^{\text {G}}) is (asymptotically) an upper bound for ρ s ( Q ) {\rho _s}(Q) , if Q is any quadrature rule with the same degree as Q n G Q_n^{\text {G}} .