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

We present a general method for construction of frames $\{\psi_I\}_{I\in \mathcal{D}}$ 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 $\theta$ such as the Gaussian $\theta(x)=\exp(-\vert x\vert^2)$. We also study the boundedness and invertibility of the frame operator $Sf=\sum_{I\in\mathcal{D}} \langle{f,\psi_I}\rangle\psi_I$ on Triebel-Lizorkin and Besov spaces and give necessary and sufficient conditions for the dual system $\{S^{-1}\psi\}_{I\in\mathcal{D}}$ to be a frame as well.