© 2017

Frames and Other Bases in Abstract and Function Spaces

Novel Methods in Harmonic Analysis, Volume 1

  • Isaac Pesenson
  • Quoc Thong Le Gia
  • Azita Mayeli
  • Hrushikesh Mhaskar
  • Ding-Xuan Zhou


  • 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


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

Table of contents

  1. Front Matter
    Pages i-xiv
  2. Introduction

    1. Front Matter
      Pages 1-1
    2. Isaac Pesenson
      Pages 3-12
  3. Frames in Abstract Spaces

    1. Front Matter
      Pages 13-13
    2. Akram Aldroubi, Armenak Petrosyan
      Pages 15-26
    3. Radu Balan, Matthew Begué, Chae Clark, Kasso Okoudjou
      Pages 27-45
    4. Travis Bemrose, Peter G. Casazza, Desai Cheng, John Haas, Hanh Van Nguyen
      Pages 81-99
  4. Space-Frequency Analysis in Function Spaces on Rn

    1. Front Matter
      Pages 125-125
    2. Giovanni S. Alberti, Stephan Dahlke, Filippo De Mari, Ernesto De Vito, Hartmut Führ
      Pages 127-160
    3. John J. Benedetto, Alfredo Nava-Tudela, Alexander M. Powell, Yang Wang
      Pages 185-213
    4. Jeffrey A. Hogan, Joseph D. Lakey
      Pages 215-235
    5. Jürgen Prestin, Christian Wülker
      Pages 237-263
    6. Holger Wendland
      Pages 265-299
  5. Frames in Spaces of Functions on Manifolds and Groups

    1. Front Matter
      Pages 301-301
    2. Swanhild Bernstein, Paul Keydel
      Pages 303-332

About this book


The first 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 I is organized around the theme of frames and other bases in abstract and function spaces, covering topics such as:
  • The advanced development of frames, including Sigma-Delta quantization for fusion frames, localization of frames, and frame conditioning, as well as applications to distributed sensor networks, Galerkin-like representation of operators, scaling on graphs, and dynamical sampling.
  • A systematic approach to shearlets with applications to wavefront sets and function spaces.
  • Prolate and generalized prolate functions, spherical Gauss-Laguerre basis functions, and radial basis functions.
  • Kernel methods, wavelets, and frames on compact and non-compact manifolds.


Sampling Frames Manifolds Time-frequency Analysis Space Frequency Data Mining

Editors and affiliations

  • Isaac Pesenson
    • 1
  • Quoc Thong Le Gia
    • 2
  • Azita Mayeli
    • 3
  • Hrushikesh Mhaskar
    • 4
  • Ding-Xuan Zhou
    • 5
  1. 1.Department of MathematicsTemple UniversityPhiladelphiaUSA
  2. 2.School of Mathematics and StatisticsThe University of New South WalesSydneyAustralia
  3. 3.Department of MathematicsThe Graduate Center, CUNYNew YorkUSA
  4. 4.Institute of Mathematical SciencesClaremont Graduate UniversityClaremontUSA
  5. 5.Department of MathematicsCity University of Hong KongKowloon TongHong Kong

Bibliographic information