Algorithms for nonlinear piecewise polynomial approximation: Theoretical aspects

In this article algorithms are developed for nonlinear -term Courant element approximation of functions in ( ) on bounded polygonal domains in . 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 rate of the best approximation and provide the basis for numerical implementation. Simple thresholding criteria enable approximation of a target function to optimally high asymptotic rates which are determined and automatically achieved by the inherent smoothness of . The algorithms provide direct approximation estimates and permit utilization of the general Jackson-Bernstein machinery to characterize -term Courant element approximation in terms of a scale of smoothness spaces (-spaces) which govern the approximation rates.

     

"Push-the-Error" Algorithm for Nonlinear n-Term Approximation

This paper is concerned with further developing and refining the analysis of a recent algorithmic paradigm for nonlinear approximation, termed the "Push-the-Error" 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 proof of the quasi-semi-additivity of the functional N(ε) counting the number of terms needed to achieve accuracy ε. This allows one to show that the improved scheme captures all rates of best n-term approximation.     

Two-Level-Split Decomposition of Anisotropic Besov Spaces

Anisotropic Besov spaces (B-spaces) are developed based on anisotropic multilevel ellipsoid covers (dilations) of ℝ ⁿ . This extends earlier results on anisotropic Besov spaces. Furthermore, sequences of anisotropic bases are constructed and utilized for two-level-split decompositions of the B-spaces and nonlinear m-term approximation.     

Localized Polynomial Frames on the Ball

Almost exponentially localized polynomial kernels are constructed on the unit ball in with weights , by smoothing out the coefficients of the corresponding orthogonal projectors. These kernels are utilized to the design of cubature formulas on with respect to and to the construction of polynomial tight frames in (called needlets) whose elements have nearly exponential localization.     

“Compactly” supported frames for spaces of distributions on the ball

Frames are constructed on the unit ball B ^d in ${\mathbb{R}^d}$ consisting of smooth functions with small shrinking supports. The new frames are designed so that they can be used for decomposition of weighted Triebel–Lizorkin and Besov spaces on B ^d with weight ${w_\mu(x):=(1-|x|^2)^{\mu-1/2}, \mu}$ half integer,?μ?≥ 0.     

Nonlinear Approximation from Dictionaries: Some Open Problems: Research Problems 2000-3

No abstract. March 31, 2000. Date accepted: March 31, 2000.     

Localized Polynomial Frames on the Interval with Jacobi Weights

As is well known the kernel of the orthogonal projector onto the polynomials of degree n in L²(w_α,β, [−1, 1]), w_α,β(t) = (1 − t)^α(1 + t)^β, can be written in terms of Jacobi polynomials. It is shown that if the coefficients in this kernel are smoothed out by sampling a compactly supported C^∞ function then the resulting function has nearly exponential (faster than any polynomial) rate of decay away from the main diagonal. This result is used for the construction of tight polynomial frames for L²(w_α,β) with elements having almost exponential localization.