New bases for Triebel-Lizorkin and Besov spaces

Abstract

We give a new method for construction of unconditional bases for general classes of Triebel-Lizorkin and Besov spaces. These include the Lp, Hp, potential, and Sobolev spaces. The main feature of our method is that the character of the basis functions can be prescribed in a very general way. In particular, if Φ is any sufficiently smooth and rapidly decaying function, then our method constructs a basis whose elements are linear combinations of a fixed (small) number of shifts and dilates of the single function Φ. Typical examples of such Φ's are the rational function Φ(·) = (1 + | · |2)-N and the Gaussian function Φ(·) = e-|miḋ|2. This paper also shows how the new bases can be utilized in nonlinear approximation.