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MAT 280 : Laplacian Eigenfunctions : Theory , Applications , and Computations Lecture 15 : Use of Laplacian Eigenfunctions and Eigenvalues for Analyzing Data on a Domain of Complicated Shape

We wish to analyze the spatial frequency inside of the object defined in Ω, at the same time, we need to avoid the Gibbs phenomenon due to the boundary Γ = ∂Ω. We also want to represent the object information efficiently for analysis, interpretation, and discrimination; this includes examining fast decaying expansion coefficients relative to a meaningful basis. Further, we want to extract geometric information about the domain Ω. For more details, see [1], [2].