Inequalities between remainders of quadratures

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.

On a certain adaptive method of approximate integration and its stopping criterion

“…Let us note that in the above proof the function 1 6 ϕ is in fact the Peano kernel of E. So, Bojanié and Roulier's result in the proof of Theorem 1 could be replaced by the Peano Kernel Theorem for 3-convex functions g ∈ C 4 [−1, 1] Then, by Bessenyei and Páles' smoothing technique (mentioned in the Introduction, cf. [1]), it is enough to extend the result to arbitrary 3convex functions g : [−1, 1] → R. This alternative approach is used in the recent paper [8] by Komisarski…”

On optimal inequalities between three-point quadratures

“…As in the previous proofs, it is enough to check the inequality 3 + dx for all a ∈ [0, 1). If a 3 4 , then…”

“…is fulfilled by any 3-convex function f : [−1, 1] → R. To find the desired condition for t it is enough to restrict ourselves to the functions (• − a) 3 + , where a ∈ [0, 1) (see Lemma 1 and the result of Bojanić and Roulier discussed just above the formulation of this lemma. Examining the proof of Lemma 2 we observe that Q z (• − a) 3 + z > 0 for z ∈ (a, 1], so the func-…”

“…Proof. Having Bojanić and Roulier's result in mind it is enough to check our assertion only for the functions (• − a) 3 + for all a ∈ (−1, 1). Observe that z → Q z (• − a) 3 + is nondecreasing if and only if the function…”

“…. , n. Hence to prove the inequality of the form (1) it is enough to check it for any spline x → (x − u) 3 + for any u ∈ (−1, 1).…”