  Numerical homogenization aims to efficiently and accurately approximate the
solution space of an elliptic partial differential operator with arbitrarily
rough coefficients in a $d$-dimensional domain. The application of the inverse
operator to some standard finite element space defines an approximation space
with uniform algebraic approximation rates with respect to the mesh size
parameter $H$. This holds even for under-resolved rough coefficients. However,
the true challenge of numerical homogenization is the localized computation of
a localized basis for such an operator-dependent approximate solution space.
This paper presents a novel localization technique that enforces the
super-exponential decay of the basis relative to $H$. This shows that basis
functions with supports of width $\mathcal O(H|\log H|^{(d-1)/d})$ are
sufficient to preserve the optimal algebraic rates of convergence in $H$
without pre-asymptotic effects. A sequence of numerical experiments illustrates
the significance of the new localization technique when compared to the so far
best localization to supports of width $\mathcal O(H|\log H|)$.
