Dany Leviatan

Learn More
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)
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 = ∞ X j=1 cj(f)φj(f), φj(f) ∈ D, j = 1, 2, . . . with the Parseval property: ‖f‖2 = j(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)
In recent years there have been various attempts at the representations of multivariate 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 estimate the degree of comonotone polynomial approximation of continuous functions f , on [−1,1], that change monotonicity s ≥ 1 times in the interval, when the degree of unconstrained polynomial approximation En(f ) ≤ n−α , n ≥ 1. We ask whether the degree of comonotone approximation is necessarily ≤ c(α, s)n−α , n ≥ 1, and if not, what can be said. It(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)
We survey developments, over the last thirty years, in the theory of Shape Preserving Approximation (SPA) by algebraic polynomials on a finite interval. In this article, “shape” refers to (finitely many changes of) monotonicity, convexity, or q-monotonicity of a function. It is rather well known that it is possible to approximate a function by algebraic(More)