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The Bramble-Hilbert lemma is a fundamental result on multivariate polynomial approximation. It is frequently applied in the analysis of Finite Element Methods (FEM) used for numerical solutions of PDEs. However, this classical estimate depends on the geometry of the domain and may 'blow-up' for simple examples such as a sequence of triangles of equivalent(More)
It is shown that certain algorithms of compression based on wavelet decompositions are optimal in the sense of nonlinear n-widths. 1. Introduction The use of wavelet decompositions for image and data compression has recently received much attention. The question arises as to whether wavelets have any significant advantage over other methods of compression(More)
Let I be a finite interval and r ∈ N. Denote by ∆ s + L q the subset of all functions y ∈ L q such that the s-difference ∆ s τ y(·) is nonnegative on I, ∀τ > 0. Further, denote by ∆ s + W r p , the class of functions x on I with the seminorm x (r) L p ≤ 1, such that ∆ s τ x ≥ 0, τ > 0. For s = 3,. .. , r + 1, we obtain two-sided estimates of the shape(More)
We are going to survey recent developments and achievements in shape preserving approximation by polynomials. We wish to approximate a function f deened on a nite interval, say ?1; 1], while preserving certain intrinsic \shape" properties. To be speciic we demand that the approximation process preserve properties of f , like its sign in all or part of the(More)
In recent years there have been various attempts at the representations of multi-variate signals such as images, which outperform wavelets. As is well known wavelets are not optimal in that they do not take full advantage of the geometrical regularities and singularities of the images. Thus these approaches have been based on tracing curves of singularities(More)
We study nonlinear m-term approximation with regard to a redundant dictionary D in a Banach space. It is known that in the case of Hilbert space H the Pure Greedy Algorithm (or, more generally, the Weak Greedy Algorithm) provides for each f ∈ H and any dictionary D an expansion into a series f = ∞ j=1 c j (f)ϕ j (f), ϕ j (f) ∈ D, j = 1, 2,. .. with the(More)
We study nonlinear m-term approximation with regard to a redundant dictionary D in a Hilbert space H. It is known that the Pure Greedy Algorithm (or, more generally, the Weak Greedy Algorithm) provides for each f ∈ H and any dictionary D an expansion into a series f = ∞ j=1 c j (f)ϕ j (f), ϕ j (f) ∈ D, j = 1, 2,. .. with the Parseval property: f 2 = j |c j(More)
The Binary Space Partition (BSP) technique is a simple and efficient method to adaptively partition an initial given domain to match the geometry of a given input function. As such the BSP technique has been widely used by practitioners, but up until now no rigorous mathematical justification to it has been offered. Here we attempt to put the technique on(More)