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

Novel Methods in Harmonic Analysis, Volume 2

  • Book
  • © 2017

Overview

Editors:
  1. Isaac Pesenson

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

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  2. Quoc Thong Le Gia

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

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  3. Azita Mayeli

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

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  4. Hrushikesh Mhaskar

    1. Institute of Mathematical Sciences, Claremont Graduate University, Claremont, USA

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  5. Ding-Xuan Zhou

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

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  • Exhibits several recently discovered links between traditional harmonic analysis and modern ideas in areas such as Riemannian geometry and sheaf theory
  • Contains both deep theoretical results and innovative applications to various fields such as medical imagine and data science
  • Only publication of its kind extending classical harmonic analysis to manifolds, graphs, and other general structures
  • Comprised of original research and survey papers from well-known experts
  • Includes supplementary material: sn.pub/extras

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

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Table of contents (16 chapters)

  1. Front Matter

    Pages i-xiv
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  3. Fourier Analysis, Its Generalizations and Applications

    1. Front Matter

      Pages 449-449
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    2. Characterization of Gevrey Regularity by a Class of FBI Transforms

      • S. Berhanu, Abraham Hailu
      Pages 451-482

    3. A Novel Mathematical Approach to the Theory of Translation Invariant Linear Systems

      • Hans G. Feichtinger
      Pages 483-516

    4. Asymptotic Behavior of the Fourier Transform of a Function of Bounded Variation

      • Elijah Liflyand
      Pages 517-532

    5. Convergence and Regularization of Sampling Series

      • W. R. Madych
      Pages 533-562

  4. Analysis on Non-Euclidean Spaces

    1. Front Matter

      Pages 563-563
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    2. Harmonic Analysis in Non-Euclidean Spaces: Theory and Application

      • Stephen D. Casey
      Pages 565-601

    3. A Harmonic Analysis of Directed Graphs from Arithmetic Functions and Primes

      • Ilwoo Cho, Palle E. T. Jorgensen
      Pages 603-651

    4. Sheaf and Duality Methods for Analyzing Multi-Model Systems

      • Michael Robinson
      Pages 653-703

  5. Harmonic Analysis and Differential Equations

    1. Front Matter

      Pages 705-705
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    2. On Boundary-Value Problems for a Partial Differential Equation with Caputo and Bessel Operators

      • Praveen Agarwal, Erkinjon Karimov, Murat Mamchuev, Michael Ruzhansky
      Pages 707-718

    3. On the Solvability of the Zaremba Problem in Infinite Sectors and the Invertibility of Associated Singular Integral Operators

      • Hussein Awala, Irina Mitrea, Katharine Ott
      Pages 719-751

    4. On the Solution of the Oblique Derivative Problem by Constructive Runge-Walsh Concepts

      • Willi Freeden, Helga Nutz
      Pages 753-794

  6. Harmonic Analysis for Data Science

    1. Front Matter

      Pages 795-795
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    2. An Overview of Numerical Acceleration Techniques for Nonlinear Dimension Reduction

      • Wojciech Czaja, Timothy Doster, Avner Halevy
      Pages 797-829

    3. Adaptive Density Estimation on the Circle by Nearly Tight Frames

      • Claudio Durastanti∗
      Pages 831-860

    4. Interactions Between Kernels, Frames, and Persistent Homology

      • Mijail Guillemard, Armin Iske
      Pages 861-888
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Keywords

About this book

The second of a two volume set on novel methods in harmonic analysis, this book draws on a number of original research and survey papers from well-known specialists detailing the latest innovations and recently discovered links between various fields. Along with many deep theoretical results, these volumes contain numerous applications to problems in signal processing, medical imaging, geodesy, statistics, and data science. 


The chapters within cover an impressive range of ideas from both traditional and modern harmonic analysis, such as: the Fourier transform, Shannon sampling, frames, wavelets, functions on Euclidean spaces, analysis on function spaces of Riemannian and sub-Riemannian manifolds, Fourier analysis on manifolds and Lie groups, analysis on combinatorial graphs, sheaves, co-sheaves, and persistent homologies on topological spaces. 


Volume II is organized around the theme of recent applications of harmonic analysis to function spaces, differential equations, and data science, covering topics such as:
  • The classical Fourier transform, the non-linear Fourier transform (FBI transform), cardinal sampling series and translation invariant linear systems.
  • Recent results concerning harmonic analysis on non-Euclidean spaces such as graphs and partially ordered sets.
  • Applications of harmonic analysis to data science and statistics
  • Boundary-value problems for PDE's  including  the Runge–Walsh theorem for the oblique derivative problem of physical geodesy.






Editors and Affiliations

  • Department of Mathematics, Temple University, Philadelphia, USA

    Isaac Pesenson

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

    Quoc Thong Le Gia

  • Department of Mathematics, The Graduate Center, CUNY, New York, USA

    Azita Mayeli

  • Institute of Mathematical Sciences, Claremont Graduate University, Claremont, USA

    Hrushikesh Mhaskar

  • Department of Mathematics, City University of Hong Kong, Kowloon Tong, Hong Kong

    Ding-Xuan Zhou

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