  In this paper, we are interested in the reiterated homogenization of linear
elliptic equations of the form $-\frac{\partial}{\partial x_{i}} \left(a_{i j}
\left(\frac{x}{\varepsilon}, \frac{x}{\varepsilon^{2}}\right) \frac{\partial
u_\varepsilon}{\partial x_{j}}\right)=f$ in $\Omega$ with Dirichlet boundary
conditions. We obtain error estimates $O(\varepsilon)$ for a bounded $C^{1,1}$
domain for this equation as well as the interior Lipschitz estimates at (very)
large scale. Compared to the general homogenization problems, the difficulty in
the reiterated homogenization is that we need to handle different scales of
$x$. To overcome this difficulty, we firstly introduce the Fourier transform in
the homogenization theory to separate these different scales. We also note that
this method may be adapted to the following reiterated homogenization problem:
$-\frac{\partial}{\partial x_{i}} \left(a_{i j}
\left(\frac{x}{\varepsilon},\cdots, \frac{x}{\varepsilon^{N}}\right)
\frac{\partial u_\varepsilon}{\partial x_{j}} \right) = f$ in $\Omega$ with
Dirichlet boundary conditions. Moreover, our results may be extended to the
related Neumann boundary problems without any real difficulty.
