  We construct smooth localized orthonormal bases compatible with homogeneous
mixed-norm Triebel-Lizorkin spaces in an anisotropic setting on $\bR^d$. The
construction is based on tensor products of so-called univariate brushlet
functions that are constructed using local trigonometric bases in the frequency
domain. It is shown that the associated decomposition system form unconditional
bases for the homogeneous mixed-norm Triebel-Lizorkin spaces.
  In the second part of the paper we study nonlinear $m$-term nonlinear
approximation with the constructed basis in the mixed-norm setting, where the
behaviour, in general, for $d\geq 2$, is shown to be fundamentally different
from the unmixed case. However, Jackson and Bernstein inequalities for $m$-term
approximation can still be derived.
