  In the whole space $R^d$, $d\ge 2$, we study homogenization of a divergence
form elliptic operator $A_\varepsilon$ of order $2m\ge 4$ with measurable
$\varepsilon$-periodic coefficients, where $\varepsilon$ is a small parameter.
For the resolvent $(A_\varepsilon+1)^{-1}$, we construct an approximation with
the remainder term of order $\varepsilon^2$ in the operator
$(L^2{\to}H^m)$-norm, using the resolvent of the homogenized operator,
solutions of several auxiliary periodic problems on the unit cube, and
smoothing operators. The homogenized operator here differs from the one
commonly employed in homogenization.
