  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}) $.
