Harmonic analysis for star graphs and the spherical coordinate trapezoidal rule

Authors:
Robert Carlson
Affiliations:
-
Venue:
Journal of Computational and Applied Mathematics
Year:
2011

Citing 3
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Abstract

Novel ideas in harmonic analysis are used to analyze the trapezoidal rule integration for two spheres. Sampling in spherical coordinates links three levels of harmonic analysis. Eigenfunctions of a nonstandard manifold Laplacian descend by restriction, first to a differential graph Laplacian, and then to difference operators. Trapezoidal rule integration with appropriate sampling is exact on eigenspaces of the manifold Laplacian, a fact which leads to trapezoidal rule error estimates on Sobolev-style spaces of functions. Singular functions with accurate trapezoidal rule integrals are identified, and a simplified analysis of smooth function numerical integration is provided.