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 ∈ ℬ γ ∑ k w k , n f ( x k , n ) − ∫ 𝕏 f d μ ∗ ≤ c n − γ , n = 1 , 2 , ⋯ , $$\displaystyle \sup _{f\in \mathcal {B}_\gamma }\left |\sum _k w_{k,n}{\,f}(x_{k,n})-\int _{\mathbb {X}} fd\mu ^*\right | \le cn^{-\gamma }, \qquad n=1,2,\cdots , $$ where 𝕏 $$\mathbb {X}$$ is often a manifold with its volume measure μ ∗, and ℬ γ $$\mathcal {B}_\gamma $$ 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 𝕏 $$\mathbb {X}$$ with a probability measure μ ∗. We show that quadrature formulas exact for integrating the so called diffusion polynomials of degree