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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.
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.
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.
This book is the first to be devoted to the theory and applications
of spherical (radial) basis functions (SBFs), which is rapidly
emerging as one of the most promising techniques for solving
problems where approximations are needed on the surface of a
sphere. The aim of the book is to provide enough theoretical and
practical details for the reader to be able to implement the SBF
methods to solve real world problems. The authors stress the close
connection between the theory of SBFs and that of the more
well-known family of radial basis functions (RBFs), which are
well-established tools for solving approximation theory problems on
more general domains. The unique solvability of the SBF
interpolation method for data fitting problems is established and
an in-depth investigation of its accuracy is provided. Two chapters
are devoted to partial differential equations (PDEs). One deals
with the practical implementation of an SBF-based solution to an
elliptic PDE and another which describes an SBF approach for
solving a parabolic time-dependent PDE, complete with error
analysis. The theory developed is illuminated with numerical
experiments throughout. Spherical Radial Basis Functions, Theory
and Applications will be of interest to graduate students and
researchers in mathematics and related fields such as the
geophysical sciences and statistics.
Volume I: http://www.springer.com/book/9783319555492 Volume II:
http://www.springer.com/book/9783319555553 A two volume set on
novel methods in harmonic analysis, these books draw 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.
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