Harmonic analysis for star graphs and the spherical coordinate trapezoidal rule

✍ Scribed by Robert Carlson

• Publisher: Elsevier Science
• Year: 2011
• Tongue: English
• Weight: 254 KB
• Volume: 235
• Category: Article
• ISSN: 0377-0427
• DOI: 10.1016/j.cam.2010.10.006

✦ Synopsis

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.