Approximate quadrature measures on data--defined spaces

Abstract

An important question in the theory of approximate integration is to study the conditions on the nodes $x_{k,n}$ and weights $w_{k,n}$ that allow an estimate of the form $$ \sup_{f\in \mathcal{B}_γ}|\sum_k w_{k,n}f(x_{k,n})-\int_\mathbb{X} fdμ^*| \le cn^{-γ}, \qquad n=1,2,\cdots, $$ where $\mathbb{X}$ is often a manifold with its volume measure $μ^*$, and $\mathcal{B}_γ$ is the unit ball of a suitably defined smoothness class, parametrized by $γ$. In this paper, we study this question in the context of a quasi-metric, locally compact, measure space $\mathbb{X}$ with a probability measure $μ^*$. We show that quadrature formulas exact for integrating the so called diffusion polynomials of degree $<n$ satisfy such estimates. Without requiring exactness, such formulas can be obtained as a solutions of some kernel-based optimization problem. We discuss the connection with the question of optimal covering radius. Our results generalize in some sense many recent results in this direction.