Approximate Quadrature Measures on Data-Defined Spaces

Hrushikesh N. Mhaskar ()
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Hrushikesh N. Mhaskar: Claremont Graduate University, Institute of Mathematical Sciences

A chapter in Contemporary Computational Mathematics - A Celebration of the 80th Birthday of Ian Sloan, 2018, pp 931-962 from Springer

Abstract: 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 ∈ 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

Keywords: Quadratic Measure; Diffusion Polynomials; Approximate Integration; Sigma Finite; Heat Kernel (search for similar items in EconPapers)
Date: 2018
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-319-72456-0_41

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DOI: 10.1007/978-3-319-72456-0_41

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