Best quadrature formulas at equally spaced nodes

Abstract

It has been known for some time that the trapezoidal rule Tnf = 1 2f(0) + f(1) + ... + f(n - 1) + 1 2f(n) is the best quadrature formula in the sense of Sard for the space W1,p, all functions such that f′ ε{lunate} Lp. In other words, the norm of the error functional Ef = ∝0n f(x) dx - ∑k = 0n λkf(k) in W1,p is uniquely minimized by the trapezoidal sum. This paper deals with quadrature formulas of the form ∑k = 0n ∑lε{lunate}J cklf(l)(k) where J is some subset of {0, 1,..., m - 1}. For certain index sets J we identify the best quadrature formula for the space Wm,p, all functions such that f(m) ε{lunate} Lp. As a result, we show that the Euler-Maclaurin quadrature formula Tnf + ∑ o<2v≤m B2v (2v)! (f (2v-1)(0) - f (2v-1) (n)) is the best quadrature formula of the above form with J = {0, 1, 3,..., ≤m - 1} for the space Wm,p, providing m is an odd integer. © 1974.