This paper is concerned with further developing and refining the analysis of a recent algorithmic paradigm for nonlinear approximation, termed the “Push-theError” scheme. It is especially designed to deal with L∞-approximation in a multilevel framework. The original version is extended considerably to cover all commonly used multiresolution frameworks. The main conceptually new result is the pr...

We study the optimal approximation of the solution of an operator equation A(u) = f by linear mappings of rank n and compare this with the best n-term approximation with respect to an optimal Riesz basis. We consider worst case errors, where f is an element of the unit ball of a Hilbert space. We apply our results to boundary value problems for elliptic PDEs that are given by an isomorphism A :...

We study the optimal approximation of the solution of an operator equation A(u) = f by linear mappings of rank n and compare this with the best n-term approximation with respect to an optimal Riesz basis. We consider worst case errors, where f is an element of the unit ball of a Hilbert space. We apply our results to boundary value problems for elliptic PDEs that are given by an isomorphism A :...

SUMMARY This paper investigates the finite sample distribution of the least squares estimator of the autoregressive parameter in a first-order autoregressive model. Uniform asymptotic expansion for the distribution applicable to both stationary and nonstationary cases is obtained. Accuracy of approximation to the distribution by a first few terms of this expansion is then investigated. It is fo...

In this article algorithms are developed for nonlinear n-term Courant element approximation of functions in Lp (0 < p ≤ ∞) on bounded polygonal domains in R2. Redundant collections of Courant elements, which are generated by multilevel nested triangulations allowing arbitrarily sharp angles, are investigated. Scalable algorithms are derived for nonlinear approximation which both capture the rat...

Nonlinear piecewise polynomial approximation: Theory and Algorithms Borislav Karaivanov We study nonlinear n-term approximation in Lp(R) (0 < p < ∞) from Courant elements or (discontinuous) piecewise polynomials generated by multilevel nested triangulations of R which allow arbitrarily sharp angles. To characterize the rate of approximation we introduce and develop three families of smoothness ...

We study N-term approximation for general families of sequence spaces, establishing sharp versions of Jackson and Bernstein inequalities. The sequence spaces used are adapted to provide characterizations of Triebel-Lizorkin and Besov spaces by means of wavelet-like systems using general dilation matrices, and thus they include spaces of anisotropic smoothness. As an application we characterize ...

We study the problem of computing the diameter of a  set of $n$ points in $d$-dimensional Euclidean space for a fixed dimension $d$, and propose a new $(1+varepsilon)$-approximation algorithm with $O(n+ 1/varepsilon^{d-1})$ time and $O(n)$ space, where $0 < varepsilonleqslant 1$. We also show that the proposed algorithm can be modified to a $(1+O(varepsilon))$-approximation algorithm with $O(n+...