This paper examines the relationship between wavelet-based image processing algorithms
and variational problems. Algorithms are derived as exact or approximate minimizers of …

This book presents a mathematical introduction to the theory of orthogonal wavelets and
their uses in analyzing functions and function spaces, both in one and in several variables …

A novel theory is introduced for analyzing image compression methods that are based on
compression of wavelet decompositions. This theory precisely relates (a) the rate of decay in …

In recent years many authors have introduced certain nonlinear algorithms for data
compression, noise removal, and image reconstruction. These methods include wavelet and …

Averaging lemmas deduce smoothness of velocity averages, such as\[\bar f (x):=\int
_\Omega f (x, v) dv,\quad\Omega\subset\mathbb {R}^ d,\] from properties of $ f $. A canonical …

We study nonlinear n-term approximation in L p (R 2)(0< p<∞) from Courant elements or
(discontinuous) piecewise polynomials generated by multilevel nested triangulations of R 2 …

We consider approximation rates of sparsely connected deep rectified linear unit (ReLU)
and rectified power unit (RePU) neural networks for functions in Besov spaces B^q(L^p) in …

We survey some recent results concerning refinable shiftinvariant spaces. These spaces
form a natural framework for the study of splines and wavelets. The following four topics will …

In this paper we introduce a polynomial frame on the unit sphere $\mathbb {S}^{d-1} $ of
$\mathbb {R}^ d $, for which every distribution has a wavelet-type decomposition. More …

We study nonlinear approximation in L p (R d)(0< p<∞, d> 1) from (a) n-term rational
functions, and (b) piecewise polynomials generated by different anisotropic dyadic partitions …