Quarklet characterizations for Triebel–Lizorkin spaces

In this paper we prove that under some conditions on the parameters the univariate Triebel–Lizorkin spaces $F_{r,q}^s\left(R\right)$ can be characterized 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 by means 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 $E_{u,r,q}^s\left(R\right)$.