New bases for Triebel-Lizorkin and Besov spaces

We give a new method for construction of unconditional bases for general classes of Triebel-Lizorkin and Besov spaces. These include the L-p, H-p, 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 Phi 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 Phi. Typical examples of such Phi 's are the rational function Phi(.) = (1 + \ , \ 2)(-N) and the Gaussian function Phi(.) = e(-2 \.\2). This paper also shows how the new bases can be utilized in nonlinear approximation.