Frames and Other Bases in Abstract and Function Spaces: Novel Methods in Harmonic Analysis, Volume 1

✍ Scribed by Isaac Pesenson, Quoc Thong Le Gia, Azita Mayeli, Hrushikesh Mhaskar, Ding-Xuan Zhou (eds.)

• Publisher: Birkhäuser Basel
• Year: 2017
• Tongue: English
• Leaves: 437
• Series: Applied and Numerical Harmonic Analysis
• Edition: 1
• Category: Library

✦ Synopsis

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.
  

✦ Table of Contents

Front Matter....Pages i-xiv
Front Matter....Pages 1-1
Frames: Theory and Practice....Pages 3-12
Front Matter....Pages 13-13
Dynamical Sampling and Systems from Iterative Actions of Operators....Pages 15-26
Optimization Methods for Frame Conditioning and Application to Graph Laplacian Scaling....Pages 27-45
A Guide to Localized Frames and Applications to Galerkin-Like Representations of Operators....Pages 47-79
Computing the Distance Between Frames and Between Subspaces of a Hilbert Space....Pages 81-99
Sigma-Delta Quantization for Fusion Frames and Distributed Sensor Networks....Pages 101-124
Front Matter....Pages 125-125
Recent Progress in Shearlet Theory: Systematic Construction of Shearlet Dilation Groups, Characterization of Wavefront Sets, and New Embeddings....Pages 127-160
Numerical Solution to an Energy Concentration Problem Associated with the Special Affine Fourier Transformation....Pages 161-184
A Frame Reconstruction Algorithm with Applications to Magnetic Resonance Imaging....Pages 185-213
Frame Properties of Shifts of Prolate and Bandpass Prolate Functions....Pages 215-235
Fast Fourier Transforms for Spherical Gauss-Laguerre Basis Functions....Pages 237-263
Multiscale Radial Basis Functions....Pages 265-299
Front Matter....Pages 301-301
Orthogonal Wavelet Frames on Manifolds Based on Conformal Mappings....Pages 303-332
Quasi Monte Carlo Integration and Kernel-Based Function Approximation on Grassmannians....Pages 333-353
Construction of Multiresolution Analysis Based on Localized Reproducing Kernels....Pages 355-375
Regular Sampling on Metabelian Nilpotent Lie Groups: The Multiplicity-Free Case....Pages 377-411
Parseval Space-Frequency Localized Frames on Sub-Riemannian Compact Homogeneous Manifolds....Pages 413-433
Back Matter....Pages 435-438

✦ Subjects

Abstract Harmonic Analysis;Fourier Analysis;Numerical Analysis;Big Data;Computational Science and Engineering