Frames and Other Bases in Abstract and Function Spaces

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The first volume of a two-part series on novel methods in harmonic analysis presents original research and survey papers from leading specialists, highlighting recent innovations and connections across various fields. It features deep theoretical results and numerous applications in signal processing, medical imaging, geodesy, statistics, and data science. The chapters explore a wide array of concepts from both traditional and modern harmonic analysis, including the Fourier transform, Shannon sampling, frames, wavelets, and functions on Euclidean spaces. Additionally, it addresses analysis on function spaces of Riemannian and sub-Riemannian manifolds, Fourier analysis on manifolds and Lie groups, as well as analysis on combinatorial graphs, sheaves, co-sheaves, and persistent homologies in topological spaces. This volume is centered on frames and bases in abstract and function spaces, discussing advanced developments in frames, such as Sigma-Delta quantization for fusion frames, localization, and conditioning, alongside applications to distributed sensor networks and dynamical sampling. It also systematically examines shearlets, prolate functions, spherical Gauss-Laguerre basis functions, and kernel methods, with a focus on wavelets and frames on both compact and non-compact manifolds.