Harmonic Analysis in Non-Euclidean Spaces: Theory and Application

We discuss harmonic analysis in the settings of both Euclidean and non-Euclidean spaces, and then focus on two specific problems using this analysis – sampling theory and network tomography. These show both the importance of non-Euclidean spaces and some of the challenges one encounters when working in non-Euclidean geometry. Starting with an overview of surfaces, we demonstrate the importance of hyperbolic space in general surface theory, and then develop harmonic analysis in general settings, looking at the Fourier-Helgason transform and its inversion. We then focus on sampling and tomography.

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