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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)

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)

Let I be a finite interval, r, n ∈ N, s ∈ N 0 and 1 ≤ p ≤ ∞. Given a set M , of functions defined on I, denote by ∆ s + M the subset of all functions y ∈ M such that the s-difference ∆ s τ y(·) is nonnegative on I, ∀τ > 0. Further, denote by W r p the Sobolev class of functions x on I with the seminorm x (r) L p ≤ 1. We obtain the exact orders of the… (More)

Let I be a nite interval, r 2 N and (t) = distft; @Ig; t 2 I. Denote by

Let f ∈ C[−1, 1] change its convexity finitely many times, in the interval. We are interested in estimating the degree of approximation of f by polynomials, and by piecewise polynomials, which are coconvex with it, namely, polynomials and piecewise polynomials that change their convexity exactly at the points where f does. We obtain Jackson type estimates… (More)

Let I be a finite interval and r, s ∈ N. Given a set M , of functions defined on I, denote by ∆ s + M the subset of all functions y ∈ M 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. Let M n (h k) := n i=1 c i h k… (More)