  This paper presents a novel multi-scale method for elliptic partial
differential equations with arbitrarily rough coefficients. In the spirit of
numerical homogenization, the method constructs problem-adapted ansatz spaces
with uniform algebraic approximation rates. Localized basis functions with the
same super-exponential localization properties as the recently proposed
Super-Localized Orthogonal Decomposition enable an efficient implementation.
The method's basis stability is enforced using a partition of unity approach. A
natural extension to higher order is presented, resulting in higher
approximation rates and enhanced localization properties. We perform a rigorous
a priori and a posteriori error analysis and confirm our theoretical findings
in a series of numerical experiments. In particular, we demonstrate the
method's applicability for challenging high-contrast channeled coefficients.
