  This paper provides maximal function characterizations of anisotropic
Triebel-Lizorkin spaces associated to general expansive matrices for the full
range of parameters $p \in (0,\infty)$, $q \in (0,\infty]$ and $\alpha \in
\mathbb{R}$. The equivalent norm is defined in terms of the decay of wavelet
coefficients, quantified by a Peetre-type space over a one-parameter dilation
group. For the Banach space regime $p,q \geq 1$, we use this characterization
to prove the existence of frames and Riesz sequences of dual molecules for the
Triebel-Lizorkin spaces; the atoms are obtained by translations and anisotropic
dilations of a single function, where neither the translation nor dilation
parameters are required to belong to a discrete subgroup. Explicit criteria for
molecules are given in terms of smoothness, decay and moment conditions.
