Regular Families of Kernels for Nonlinear Approximation

Abstract

This article studies sufficient conditions on families of approximating kernels which provide $N$--term approximation errors from an associated nonlinear approximation space which match the best known orders of $N$--term wavelet expansion. These conditions provide a framework which encompasses some notable approximation kernels including splines, so-called cardinal functions, and many radial basis functions such as the Gaussians and general multiquadrics. Examples of such kernels are given to justify the criteria, and some computational experiments are done to demonstrate the theoretical results. Additionally, the techniques involved allow for some new results on $N$--term interpolation of Sobolev functions via radial basis functions.

References in corpus (7)

• On the Convergence of Regular Families of Cardinal Interpolators
• Cardinal Interpolation With General Multiquadrics
• Approximation Rates for Interpolation of Sobolev Functions via Gaussians and Allied Functions
• Cardinal Interpolation With General Multiquadrics: Convergence Rates
• p-Riesz bases in quasi shift invariant spaces
• Convergence properties of spline-like cardinal interpolation operators acting on $l^p$ data
• Compression using Quasi-Interpolation

Cited by in corpus (2)

• A Solution for the Greedy Approximation of a Step Function with a Waveform Dictionary
• Nonuniform sampling and approximation in Sobolev space from the perturbation of framelet system