Approximate quadrature measures on data--defined spaces

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 $$\sup_{f\in \mathcal{B}_\gamma}|\sum_k w_{k,n}f(x_{k,n})-\int_\mathbb{X} fd\mu^*| \le cn^{-\gamma}, \qquad n=1,2,\cdots,$$ where $\mathbb{X}$ is often a manifold with its volume measure $\mu^*$, and $\mathcal{B}_\gamma$ is the unit ball of a suitably defined smoothness class, parametrized by $\gamma$. In this paper, we study this question in the context of a quasi-metric, locally compact, measure space $\mathbb{X}$ with a probability measure $\mu^*$. 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.