On the Construction of Bases and Frames with Applications

Abstract

This article is a survey of results on the construction of bases and frames for Besov and Triebel-Lizorkin spaces (B- and F-spaces) in various classical and nonclassical settings.We first review our small-perturbation method for construction of bases for B- and F-spaces on $\mathbb{R}^{\textit{d}}$ based on wavelets and show some of its applications to nonlinear approximation. Then we describe the construction of frames for B- and F-spaces in the classical setting on $\mathbb{R}^{\textit{d}}$, on the sphere in $\mathbb{R}^{\textit{d}}$, and in a general setting that covers a number of classical and nonclassical setups such as on $\mathbb{R}^{\textit{d}}$, the interval, sphere, ball, simplex as well as Riemannian manifolds and Lee groups. Finally, we present a small perturbation method for construction of frames for Band F-spaces in a general setting. Several applications of this scheme are presented.