Harmonic analysis for star graphs and the spherical coordinate trapezoidal rule |
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Authors: | Robert Carlson |
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Institution: | Department of Mathematics, University of Colorado at Colorado Springs, Colorado Springs, CO 80921, United States |
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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. |
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Keywords: | 65D30 65T99 34B45 |
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