Barycentric Covariance and Variance-Based Quadrature Inequalities on Timescales

In this paper, we develop a variance-based framework for the study of quadrature errors on arbitrary timescales. We introduce the timescale barycenter and the associated geometric variance, which allow the construction of a covariance operator adapted to the underlying timescale geometry. Using these notions, we establish a barycentric quadrature representation formula showing that the quadrature error can be expressed as a covariance between the independent variable and the delta derivative. As a consequence, we derive a fundamental variance inequality that bounds the barycentric quadrature error in terms of the geometric dispersion of the timescale and the variance of the delta derivative. The inequality is shown to be sharp, and a complete characterization of the equality case is obtained. Applications to the continuous, discrete, and quantum timescales are presented, illustrating the flexibility of the proposed framework and its ability to accommodate both homogeneous and non-homogeneous geometries.