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 $\Phi (\cdot) = (1 + \vert\cdot\vert^2)^{-N}$ and the Gaussian function $\Phi (\cdot) = e^{-\vert\cdot\vert^2}.$ This paper also shows how the new bases can be utilized in nonlinear approximation.