On the construction of frames for Triebel-Lizorkin and Besov spaces

Abstract

We present a general method for construction of frames {ψ I} I∈D for Triebel-Lizorkin and Besov spaces, whose nature can be prescribed. In particular, our method allows for constructing frames consisting of rational functions or more general functions which are linear combinations of a fixed (small) number of shifts and dilates of a single smooth and rapidly decaying function θ such as the Gaussian θ(x) = exp(-|x| 2). We also study the boundedness and invertibility of the frame operator Sf = ∑ I∈D 〈f, ψ I〉ψ I on Triebel-Lizorkin and Besov spaces and give necessary and sufficient conditions for the dual system {S -1ψ} I∈D to be a frame as well. ©2005 American Mathematical Society.