  Let $\ell\in\mathbb{N}$ and $p\in(1,\infty]$. In this article, the authors
prove that the sequence $\{f-B_{\ell,2^{-k}}f\}_{k\in\mathbb{Z}}$ consisting of
the differences between $f$ and the ball average $B_{\ell,2^{-k}}f$
characterizes the Besov space $\dot B^\alpha_{p,q}(\rn)$ with $q\in (0,
\infty]$ and the Triebel-Lizorkin space $\dot F^\alpha_{p,q}(\rn)$ with $q\in
(1,\infty]$ when the smoothness order $\alpha\in(0,2\ell)$. More precisely, it
is proved that $f-B_{\ell,2^{-k}}f$ plays the same role as the approximation to
the identity $\varphi_{2^{-k}}\ast f$ appearing in the definitions of $\dot
B^\alpha_{p,q}(\rn)$ and $\dot F^\alpha_{p,q}(\rn)$. The corresponding results
for inhomogeneous Besov and Triebel-Lizorkin spaces are also obtained. These
results, for the first time, give a way to introduce Besov and Triebel-Lizorkin
spaces with any smoothness order in $(0, 2\ell)$ on spaces of homogeneous type,
where $\ell\in{\mathbb N}$.
