Spectral Laplace Transform of Signals on Arbitrary Domains

Abstract The Laplace transform is a central mathematical tool for analysing 1D/2D signals and the solution to PDEs; however, its definition computation on arbitrary data still an open research problem. We introduce domains focus spectral , which defined by applying 1D filter Laplacian spectrum of input domain. satisfies standard properties 2D transforms, such as dilation, translation, scaling, derivation, localisation, relations with Fourier transform. enough general be applied different discrete data, graphs, 3D surface meshes, point sets. Working in domain polynomial rational approximations, we achieve stable Transform. As main applications, discuss Transform functions (e.g., surfaces, graphs), heat diffusion equation, graph signal processing.