Quarklet Characterizations for Triebel-Lizorkin Spaces

In this paper we prove that under some conditions on the parameters the one-dimensional Triebel-Lizorkin spaces $F^{s}_{r,q}(\mathbb{R})$ can be described in terms of quarklets. So for functions from Triebel-Lizorkin spaces we obtain a quarkonial decomposition as well as a new equivalent quasi-norm. For that purpose we use quarklets that are constructed out of biorthogonal compactly supported Cohen-Daubechies-Feauveau spline wavelets, where the primal generator is a cardinal B-spline. Moreover we introduce some sequence spaces apposite to our quarklet system and study their properties. Finally we also obtain a quarklet characterization for the Triebel-Lizorkin-Morrey spaces $\mathcal{E}^{s}_{u,r,q}(\mathbb{R})$.