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Harmonic Analysis in Non-Euclidean Spaces: Theory and Application

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Recent Applications of Harmonic Analysis to Function Spaces, Differential Equations, and Data Science

Part of the book series: Applied and Numerical Harmonic Analysis ((ANHA))

Abstract

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.

Sampling theory is a fundamental area of study in harmonic analysis and signal and image processing. We connect sampling theory with the geometry of the signal and its domain. It is relatively easy to demonstrate this connection in Euclidean spaces, but one quickly gets into open problems when the underlying space is not Euclidean. We discuss how to extend this connection to hyperbolic geometry and general surfaces, outlining an Erlangen-type program for sampling theory.

The second problem we discuss is network tomography. We demonstrate a way to create a system that will detect viruses as early as possible and work simply on the geometry or structure of the network itself. Our analysis looks at weighted graphs and how the weights change due to an increase in traffic. The analysis is developed by applying the tools of harmonic analysis in hyperbolic space.

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Acknowledgements

Author’s research was partially supported by US Army Research Office Scientific Services program, administered by Battelle (TCN 06150, Contract DAAD19-02-D-0001) and US Air Force Office of Scientific Research Grant Number FA9550-12-1-0430. The author would also like to thank the referees for suggestions that helped improve the paper, his colleague Jens Christensen for conversations relevant to the paper, and his students Danielle Beard, Jackson Williams, and Emma Zaballos for studying various components of the research in the paper.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, American University, Washington, DC, 20016-8050, USA

    Stephen D. Casey

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  1. Stephen D. Casey
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Correspondence to Stephen D. Casey .

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Editors and Affiliations

  1. Department of Mathematics, Temple University, Philadelphia, Pennsylvania, USA

    Isaac Pesenson

  2. School of Mathematics and Statistics, University of New South Wales, Sydney, New South Wales, Australia

    Quoc Thong Le Gia

  3. Department of Mathematics, The Graduate Center, CUNY, New York, New York, USA

    Azita Mayeli

  4. Institute of Mathematical Sciences, Claremont Graduate University, Claremont, California, USA

    Hrushikesh Mhaskar

  5. Department of Mathematics, City University of Hong Kong, Kowloon Tong, Hong Kong

    Ding-Xuan Zhou

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Casey, S.D. (2017). Harmonic Analysis in Non-Euclidean Spaces: Theory and Application. In: Pesenson, I., Le Gia, Q., Mayeli, A., Mhaskar, H., Zhou, DX. (eds) Recent Applications of Harmonic Analysis to Function Spaces, Differential Equations, and Data Science. Applied and Numerical Harmonic Analysis. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-55556-0_6

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