# Two-Point Quadrature Rules for Riemann-Stieltjes Integrals with Lp-error estimates

## TL;DR

This paper introduces new two-point quadrature rules for Riemann-Stieltjes integrals, providing sharp $L^p$ error estimates under Hölder and bounded variation conditions, enhancing numerical integration accuracy.

## Contribution

The paper develops a general class of two-point quadrature rules for Riemann-Stieltjes integrals with proven dual formulas and sharp $L^p$ error bounds, advancing numerical integration methods.

## Key findings

- Derived new two-point quadrature rules for Riemann-Stieltjes integrals.
- Established sharp $L^p$ error estimates for the proposed rules.
- Proved dual formulas under Hölder and bounded variation assumptions.

## Abstract

In this work, we construct a new general two-point quadratre rules for the Riemann--Stieltjes integral $\int_a^b {f\left( t \right)du\left( t \right)}$, where the integrand $f$ is assumed to be satisfied with the H\"{o}lder condition on $[a,b]$ and the integrator $u$ is of bounded variation on $[a,b]$. The dual formulas under the same assumption are proved. Some sharp error $L^p$--Error estimates for the proposed quadrature rules are also obtained.