Since their introduction in the 1980's, wavelets have become a
powerful tool in mathematical analysis, with applications such as
image compression, statistical estimation and numerical simulation
of partial differential equations. One of their main attractive
features is the ability to accurately represent fairly general
functions with a small number of adaptively chosen wavelet
coefficients, as well as to characterize the smoothness of such
functions from the numerical behaviour of these coefficients. The
theoretical pillar that underlies such properties involves
approximation theory and function spaces, and plays a pivotal role
in the analysis of wavelet-based numerical methods.
This book offers a self-contained treatment of wavelets, which
includes this theoretical pillar and it applications to the
numerical treatment of partial differential equations. Its key
features are:
1. Self-contained introduction to wavelet bases and related
numerical algorithms, from the simplest examples to the most
numerically useful general constructions.
2. Full treatment of the theoretical foundations that are crucial
for the analysis
of wavelets and other related multiscale methods: function spaces,
linear and nonlinear approximation, interpolation theory.
3. Applications of these concepts to the numerical treatment of
partial differential equations: multilevel preconditioning, sparse
approximations of differential and integral operators, adaptive
discretization strategies.
General
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