Inequalities between remainders of quadratures

TL;DR

This paper investigates inequalities between the remainders of various quadrature rules for convex functions of higher orders, establishing bounds and non-negativity results using the Peano Kernel Theorem.

Contribution

It extends known inequalities for convex functions to higher-order convex functions, analyzing remainders of specific quadratures and providing new bounds and counterexamples.

Findings

Remainder of 2-point Gauss quadrature is non-negative for 3-convex functions.

Remainder of 3-point Gauss quadrature is bounded by Simpson's rule for 5-convex functions.

Counterexamples show similar bounds do not hold for even-order convex functions.

Abstract

It is well-known that in the class of convex functions the (nonnegative) remainder of the Midpoint Rule of the approximate integration is majorized by the remainder of the Trapezoid Rule. Hence the approximation of the integral of the convex function by the Midpoint Rule is better than the analogous approximation by the Trapezoid Rule. Following this fact we examine remainders of certain quadratures in the classes of convex functions of higher orders. Our main results state that for 3-convex (5-convex, respectively) functions the remainder of the 2-point (3-point, respectively) Gauss quadrature is non-negative and it is not greater than the remainder of the Simpson's Rule (4-point Lobatto quadrature, respectively). We also check the 2-point Radau quadratures for 2-convex functions to demonstrate that similar results fail to hold for convex functions of even orders. We apply Peano Kernel Theorem as a~main tool of our considerations.