How sharp are error bounds? --lower bounds on quadrature worst-case errors for analytic functions

TL;DR

This paper investigates the sharpness of error bounds in numerical integration of analytic functions, deriving lower bounds that closely match upper bounds for various quadrature rules, and revealing new insights into their optimality.

Contribution

The paper introduces novel lower bounds for quadrature errors, improves existing bounds for the trapezoidal rule, and analyzes the optimality of Gauss--Hermite, Gauss--Legendre, and Clenshaw--Curtis quadratures.

Findings

Lower bounds for quadrature errors are established, matching upper bounds closely.

The trapezoidal rule's worst-case error cannot be improved beyond a polynomial factor.

Gauss--Hermite quadrature is shown to be sub-optimal under certain decay conditions.

Abstract

Numerical integration over the real line for analytic functions is studied. Our main focus is on the sharpness of the error bounds. We first derive two general lower estimates for the worst-case integration error, and then apply these to establish lower bounds for various quadrature rules. These bounds turn out to be either novel or improve upon existing results, leading to lower bounds that closely match upper bounds for various formulas. Specifically, for the suitably truncated trapezoidal rule, we improve upon general lower bounds on the worst-case error obtained by Sugihara [\textit{Numer. Math.}, 75 (1997), pp.~379--395] and provide exceptionally sharp lower bounds apart from a polynomial factor, and in particular show that the worst-case error for the trapezoidal rule by Sugihara is not improvable by more than a polynomial factor. Additionally, our research reveals a discrepancy between the error decay of the trapezoidal rule and Sugihara's lower bound for general numerical integration rules, introducing a new open problem. Moreover, Gauss--Hermite quadrature is proven sub-optimal under the decay conditions on integrands we consider, a result not deducible from upper-bound arguments alone. Furthermore, to establish the near-optimality of the suitably scaled Gauss--Legendre and Clenshaw--Curtis quadratures, we generalize a recent result of Trefethen [\textit{SIAM Rev.}, 64 (2022), pp.~132--150] for the upper error bounds in terms of the decay conditions.