Asymptotically optimal definite quadrature formulae of $4$-th order

TL;DR

This paper develops sequences of fourth-order quadrature formulas that are asymptotically optimal, with explicit weights and nodes, and provides error estimates and monotonicity properties for convex functions.

Contribution

It introduces new asymptotically optimal quadrature formulas of fourth order with explicit weights and nodes, along with error analysis and monotonicity results.

Findings

Formulas are asymptotically optimal of fourth order.

Explicit weights and nodes are provided.

Error estimates and monotonicity properties are established.

Abstract

We construct several sequences of asymptotically optimal definite quadrature formulae of fourth order and evaluate their error constants. Besides the asymptotical optimality, an advantage of our quadrature formulae is the explicit form of their weights and nodes. For the remainders of our quadrature formulae monotonicity properties are established when the integrand is a 4-convex function, and a-posteriori error estimates are proven.