MATHEMATICS OF COMPUTATION, VOLUME 26, NUMBER 120, OCTOBER 1972 How Slowly Can Quadrature Formulas Converge? Abstract. Let {Q„)denote a sequenceof quadrature formulas, Q„(j) m Yfj-iW^fix^), such that ß„(/) -> P0 j(x) dx for all / G CTO, 1], Let 0 < e < \ and a sequence (aX_j.be given, where a, ä si ^ a, 5 • • • , and where a„ —> 0 as n —*c°. Then there exists a function / G CTO,l]and a sequence |nt-)"=i suchthat |/(x)| g 2(7,71(1 - 4e)|, and such that n,Kx)dx - Q„k(1) = ak,k = 1,2, 3, ••• . 1. Introduction and Statement of Results. We consider a sequence of quad- rature formulas !ß„j,?-i defined by where {^„1^-1 is a sequence of (increasing) positive integers and 0 i£ xjn) ^ 1 for all n and _/. The quadrature formulas, we assume, are such that for all functions / that are continuous on the closed interval [0, 1]; that is, for all / in C[0, 1]. For example, the Gaussian quadrature formulas and the well-known trapezoidal formulas have these properties. In this paper, we show that no matter what the sequence \Qn\7-i defined by (1.1) and (1.2) is, there is a function / in C[0, 1] for which {ß„/ir_i converges to // very slowly. That is, the assumption of continuity is not enough to insure the rapid convergence of any quadrature scheme. More precisely, our main result is the following: Theorem 1. Let a sequence of quadrature formulas \Qn\7-i defined by (1.1) and satisfying (1.2) for all / in C[0, 1] be given, and let \an\ be any sequence of numbers such that (1.3) lim an = 0, By Peter R. Lipow and Frank Stenger (1.1) oj = z H-:">/(*!n)) (1.2) and (1.4) a„+i I < °° . Received February 16, 1972. AMS 1970 subject classifications. Primary 65D30. Key words and phrases. Quadrature rules, convergence. Copyright © 1972, American Mathematical Society 917 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use