  In this paper, the authors establish new characterizations of the recently
introduced Besov-type spaces $\dot{B}^{s,\tau}_{p,q}({\mathbb R}^n)$ and
Triebel-Lizorkin-type spaces $\dot{F}^{s,\tau}_{p,q}({\mathbb R}^n)$ with $p\in
(0,\infty]$, $s\in{\mathbb R}$, $\tau\in [0,\infty)$, and $q\in (0,\infty]$, as
well as their preduals, the Besov-Hausdorff spaces
$B\dot{H}^{s,\tau}_{p,q}(\R^n)$ and Triebel-Lizorkin-Hausdorff spaces
$F\dot{H}^{s,\tau}_{p,q}(\R^n)$, in terms of the local means, the Peetre
maximal function of local means, and the tent space (the Lusin area function)
in both discrete and continuous types. As applications, the authors then obtain
interpretations as coorbits in the sense of H. Rauhut in [Studia Math. 180
(2007), 237-253] and discretizations via the biorthogonal wavelet bases for the
full range of parameters of these function spaces. Even for some special cases
of this setting such as $\dot F^s_{\infty,q}({\mathbb R}^n)$ for $s\in{\mathbb
R}$, $q\in (0,\infty]$ (including $\mathop\mathrm{BMO} ({\mathbb R}^n)$ when
$s=0$, $q=2$), the $Q$ space $Q_\alpha ({\mathbb R}^n)$, the Hardy-Hausdorff
space $HH_{-\alpha}({\mathbb R}^n)$ for $\alpha\in (0,\min\{\frac n2,1\})$, the
Morrey space ${\mathcal M}^u_p({\mathbb R}^n)$ for $1<p\le u<\infty$, and the
Triebel-Lizorkin-Morrey space $\dot{\mathcal{E}}^s_{upq}({\mathbb R}^n)$ for
$0<p\le u<\infty$, $s\in{\mathbb R}$ and $q\in(0,\infty]$, some of these
results are new.
