Frames and Other Bases in Abstract and Function Spaces

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.

  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

  Inhalt
  Frames: Theory and Practice.- Dynamical Sampling and Systems from Iterative Actions of Operators.- Optimization Methods for Frame Conditioning and Application to Graph Laplacian Scaling.- A Guide to Localized Frames and Applications to Galerkin-Like Representations of Operators.- Computing the Distance between Frames and between Subspaces of a Hilbert Space.- Sigma-Delta Quantization for Fusion Frames and Distributed Sensor Networks.- Recent Progress in Shearlet Theory: Systematic Construction of Shearlet Dilation Groups, Characterization of Wavefront Sets, and New Embeddings.- Numerical Solution to an Energy Concentration Problem Associated with the Special Affine Fourier Transformation.- A Frame Reconstruction Algorithm with Applications to Magnetic Resonance Imaging.- Frame Properties of Shifts of Prolate and Bandpass Prolate Functions.- Fast Fourier Transforms for Spherical Gauss-Laguerre Basis Functions.- Multiscale Radial Basis Functions.- Orthogonal Wavelet Frames on Manifolds Based on Conformal Mappings.- Quasi Monte Carlo Integration and Kernel-Based Function Approximation on Grassmannians.- Construction of Multiresolution Analysis Based on Localized Reproducing Kernels.- Regular Sampling on Metabelian Nilpotent Lie Groups: The Multiplicity-Free Case.- Parseval Space-Frequency Localized Frames on Sub-Riemann Compact Homogeneous Manifolds.<p